Research Article A Study of I-Function of Several Complex Variables Prathima Jayarama, 1,2 Vasudevan Nambisan Theke Madam, 3 and Shantha Kumari Kurumujji 2,4 1 Department of Mathematics, Manipal Institute of Technology, Manipal, Karnataka 576104, India 2 SCSVMV, Sri Jayendra Saraswathi Street, Enathur, Kanchipuram, Tamil Nadu 631561, India 3 Department of Mathematics, College of Engineering, Trikaripur, Kerala 670307, India 4 Department of Mathematics, P.A. College of Engineering, Mangalore, Karnataka 574153, India Correspondence should be addressed to Prathima Jayarama; [email protected] Received 27 June 2013; Revised 5 September 2013; Accepted 23 September 2013; Published 27 January 2014 Academic Editor: Alberto Cardona Copyright © 2014 Prathima Jayarama et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e aim of this paper is to introduce a natural generalization of the well-known, interesting, and useful Fox H-function into generalized function of several variables, namely, the I-function of “” variables. For =1, we get the I-function introduced and studied by Arjun Rathie (1997) and, for =2, we get I-function of two variables introduced very recently by ShanthaKumari et al. (2012). Convergent conditions, elementary properties, and special cases have also been given. e results presented in this paper generalize the results of H-function of “” variables available in the literature. 1. Introduction In 1997, Rathie introduced the generalization of the well- known Fox’s H-function [1] which has very recently found interesting applications in wireless communication [2–4]. Motivated by the I-function, very recently Shantha Kumari, Nambisan, and Rathie introduced I-function of two variables [5] which is a natural generalization of the H-function of two variables introduced earlier by Mittal and Gupta [6] and discussed some of its important properties. In the present paper, we aim to develop I-function of “” variables which may be regarded as the natural generalization of the H-function of “” variables introduced earlier by Srivastava and Panda [7]. We also discussed some of the important properties. e remainder of this paper is organized as follows. In Section 2, we have defined the I-function of “” variables by means of multiple Mellin-Barnes type contour integrals. In Section 3, we have given the convergence con- ditions for this function. In Section 4, we obtained the series representation and behaviour of the function for small values of the variables. In Section 5, we have mentioned special cases of our function giving relations with other functions available in the literature. Finally, in Section 6, we have mentioned a few important properties. 2. The I-Function of Several Variables e generalized Fox H-function, namely, I-function of “” variables, is defined and represented in the following manner: I [ 1 ,..., ]= I 0,: 1 , 1 ;...; , ,: 1 , 1 ;...; , [ [ [ 1 . . . ( ; (1) ,..., () ; ) 1, : ( (1) , (1) ; (1) ) 1, 1 ;...;( () , () ; () ) 1, ( ; (1) ,..., () ; ) 1, : ( (1) , (1) ; (1) ) 1, 1 ;...;( () , () ; () ) 1, ] ] ] = 1 (2) ∫ L 1 ⋅⋅⋅∫ L ( 1 ,..., ) 1 ( 1 )⋅⋅⋅ ( ) 1 1 ⋅⋅⋅ 1 ⋅ ⋅ ⋅ , (1) Hindawi Publishing Corporation International Journal of Engineering Mathematics Volume 2014, Article ID 931395, 11 pages http://dx.doi.org/10.1155/2014/931395
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Research ArticleA Study of I-Function of Several Complex Variables
Prathima Jayarama12 Vasudevan Nambisan Theke Madam3
and Shantha Kumari Kurumujji24
1 Department of Mathematics Manipal Institute of Technology Manipal Karnataka 576104 India2 SCSVMV Sri Jayendra Saraswathi Street Enathur Kanchipuram Tamil Nadu 631561 India3 Department of Mathematics College of Engineering Trikaripur Kerala 670307 India4Department of Mathematics PA College of Engineering Mangalore Karnataka 574153 India
Correspondence should be addressed to Prathima Jayarama pamrutharajyahoocoin
Received 27 June 2013 Revised 5 September 2013 Accepted 23 September 2013 Published 27 January 2014
Academic Editor Alberto Cardona
Copyright copy 2014 Prathima Jayarama et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
The aim of this paper is to introduce a natural generalization of the well-known interesting and useful Fox H-function intogeneralized function of several variables namely the I-function of ldquo119903rdquo variables For 119903 = 1 we get the I-function introduced andstudied by Arjun Rathie (1997) and for 119903 = 2 we get I-function of two variables introduced very recently by ShanthaKumari et al(2012) Convergent conditions elementary properties and special cases have also been given The results presented in this papergeneralize the results of H-function of ldquo119903rdquo variables available in the literature
1 Introduction
In 1997 Rathie introduced the generalization of the well-known Foxrsquos H-function [1] which has very recently foundinteresting applications in wireless communication [2ndash4]Motivated by the I-function very recently Shantha KumariNambisan and Rathie introduced I-function of two variables[5] which is a natural generalization of the H-function oftwo variables introduced earlier by Mittal and Gupta [6] anddiscussed some of its important properties
In the present paper we aim to develop I-function of ldquo119903rdquovariables whichmay be regarded as the natural generalizationof the H-function of ldquo119903rdquo variables introduced earlier bySrivastava and Panda [7] We also discussed some of theimportant properties
The remainder of this paper is organized as follows
In Section 2 we have defined the I-function of ldquo119903rdquovariables by means of multiple Mellin-Barnes type contourintegrals In Section 3 we have given the convergence con-ditions for this function In Section 4 we obtained the seriesrepresentation and behaviour of the function for small valuesof the variables In Section 5 we havementioned special casesof our function giving relations with other functions availablein the literature Finally in Section 6 we have mentioned afew important properties
2 The I-Function of Several Variables
The generalized Fox H-function namely I-function of ldquo119903rdquovariables is defined and represented in the followingmanner
I [1199111 119911
119903] = I011989911989811198991119898119903119899119903
11990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
(2120587119894)119903int
L1
sdot sdot sdot int
L119903
120601 (1199041 119904
119903) 120579
1(1199041) sdot sdot sdot 120579
119903(119904119903) 119911
1199041
1sdot sdot sdot 119911
119904119903
119903119889119904
1sdot sdot sdot 119889119904
119903
(1)
Hindawi Publishing CorporationInternational Journal of Engineering MathematicsVolume 2014 Article ID 931395 11 pageshttpdxdoiorg1011552014931395
2 International Journal of Engineering Mathematics
where 120601(1199041 119904
119903) 120579
119894(119904119894) 119894 = 1 119903 are given by
120601 (1199041 119904
119903) =
119899
prod
119895=1
Γ119860119895(1 minus 119886
119895+
119903
sum
119894=1
120572(119894)
119895119904119894)
times (
119901
prod
119895=119899+1
Γ119860119895(119886
119895minus
119903
sum
119894=1
120572(119894)
119895119904119894)
times
119902
prod
119895=1
Γ119861119895(1 minus 119887
119895+
119903
sum
119894=1
120573(119894)
119895119904119894))
minus1
(2)
120579119894(119904119894) = (
119899119894
prod
119895=1
Γ119862(119894)
119895(1 minus 119888
(119894)
119895+ 120574
(119894)
119895119904119894)
times
119898119894
prod
119895=1
Γ119863(119894)
119895(119889
(119894)
119895minus 120575
(119894)
119895119904119894))
times (
119901119894
prod
119895=119899119894+1
Γ119862(119894)
119895(119888
(119894)
119895minus 120574
(119894)
119895119904119894)
times
119902119894
prod
119895=119898119894+1
Γ119863(119894)
119895(1 minus 119889
(119894)
119895+ 120575
(119894)
119895119904119894))
minus1
(3)
where 119894 = 1 119903Also
(i) 119911119894
= 0 for 119894 = 1 119903(ii) 119894 = radicminus1(iii) an empty product is interpreted as unity(iv) the parameters 119898
119895 119899
119895 119901
119895 119902
119895(119895 = 1 119903) 119899 119901 and
119902 are nonnegative integers such that 0 le 119899 le 119901 119902 ge 00 le 119899
119895le 119901
119895 and 0 le 119898
119895le 119902
119895(119895 = 1 119903) (not all
zero simulataneously)(v) 120572
(119894)
119895(119895 = 1 119901 119894 = 1 119903) 120573(119894)
119895(119895 = 1 119902 119894 =
1 119903) 120574(119894)119895
(119895 = 1 119901119894 119894 = 1 119903) and 120575
(119894)
119895(119895 =
1 119902119894 119894 = 1 119903) are assumed to be positive
quantities for standardisation purpose However thedefinition of I-function of ldquo119903rdquo variables will have ameaning even if some of the quantities are zero ornegative numbers For these we may obtain cor-responding transformation formulas which will begiven in a later section
(vi) 119886119895(119895 = 1 119901) 119887
119895(119895 = 1 119902) 119888(119894)
119895(119895 = 1 119901
119894
119894 = 1 119903) and 119889(119894)
119895(119895 = 1 119902
119894 119894 = 1 119903) are
complex numbers(vii) the exponents 119860
119895(119895 = 1 119901) 119861
119895(119895 = 1 119902)
119862(119894)
119895(119895 = 1 119901
119894 119894 = 1 119903) and 119863
(119894)
119895(119895 =
1 119902119894 119894 = 1 119903) of various gamma functions
involved in (2) and (3) may take noninteger values(viii) the contour L
119894in the complex 119904
119894-plane is of Mellin-
Barnes type which runs from 119888minus 119894infin to 119888+ 119894infin (119888 real)with indentation if necessary in such a manner thatall singularities of Γ119863
(119894)
119895(119889
(119894)
119895minus 120575
(119894)
119895119904119894) 119895 = 1 119898
119894lie
to the right and Γ119862(119894)
119895(1 minus 119888
(119894)
119895+ 120574
(119894)
119895119904119894) 119895 = 1 119899
119894are
to the left ofL119894
Following the results of Braaksma [8] the I-function of ldquo119903rdquovariables is analytic if
120583119894=
119901
sum
119895=1
119860119895120572(119894)
119895minus
119902
sum
119895=1
119861119895120573(119894)
119895+
119901119894
sum
119895=1
119862(119894)
119895120574(119894)
119895
minus
119902119894
sum
119895=1
119863(119894)
119895120575(119894)
119895le 0 119894 = 1 119903
(4)
3 Convergence Conditions
Integral (1) converges absolutely if
1003816100381610038161003816arg (119911
119896)1003816100381610038161003816lt
1
2
Δ119896120587 119896 = 1 119903 (5)
where
Δ119896=
[
[
minus
119901
sum
119895=119899+1
119860119895120572(119896)
119895minus
119902
sum
119895=1
119861119895120573(119896)
119895
+
119898119896
sum
1
119863(119896)
119895120575(119896)
119895minus
119902119896
sum
119898119896+1
119863(119896)
119895120575(119896)
119895
+
119899119896
sum
119895=1
119862(119896)
119895120574(119896)
119895minus
119901119896
sum
119895=119899119896+1
119862(119896)
119895120574(119896)
119895]
]
gt 0
(6)
and if | arg(119911119896)| = (12)Δ
119896120587 and Δ
119896ge 0 119896 = 1 119903
then integral (1) converges absolutely under the followingconditions
(i) 120583119896= 0 Ω
119896lt minus1 where 120583
119896is given by (4) and
Ω119896=
119901
sum
119895=1
[
1
2
minus R (119886119895)]119860
119895
minus
119902
sum
119895=1
[
1
2
minus R (119887119895)] 119861
119895
+
119901119896
sum
119895=1
[
1
2
minus R (119888(119896)
119895)]119862
(119896)
119895
minus
119902119896
sum
119895=1
[
1
2
minus R (119889(119896)
119895)]119863
(119896)
119895
119896 = 1 119903
(7)
(ii) 120583119896
= 0 with 119904119896
= 120590119896+ 119894119905
119896 (120590
119896and 119905
119896are real 119896 =
1 119903) and 120590119896are chosen so that for |119905
119896| rarr infin we
have Ω119896+ 120590
119896120583119896lt minus1
Outline of the ProofThe convergence of integral (1) dependson the asymptotic behaviour of the functions 120601(119904
1 119904
119903)
120579119894(119904119894) 119894 = 1 119903 defined by (2) and (3) respectively Such
International Journal of Engineering Mathematics 3
asymptotic behaviour is based on the following relation forthe gamma function Γ(119911) 119911 = 119909 + 119894119910 119909 119910 isin R [9]
1003816100381610038161003816Γ (119909 + 119894119910)
1003816100381610038161003816sim radic2120587
10038161003816100381610038161199101003816100381610038161003816
119909minus12 exp(minus
1
2
12058710038161003816100381610038161199101003816100381610038161003816)
10038161003816100381610038161199101003816100381610038161003816rarr infin
(8)
Along the contour L119896 if we put 119904
119896= 120590
119896+ 119894119905
119896and take the
limit as |119905119896| rarr infin for 119896 = 1 119903 we obtain by virtue of (8)
that1003816100381610038161003816100381610038161003816100381610038161003816
Γ119860119895(1 minus 119886
119895+
119903
sum
119896=1
120572(119896)
119895119904119896)
1003816100381610038161003816100381610038161003816100381610038161003816
le (2120587)1198601198952(120572
(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816)
[12minusR(119886119895)+120572(119896)
119895120590119896]119860119895
times exp [minus
120587
2
(120572(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816+
10038161003816100381610038161003816I (119886
119895)
10038161003816100381610038161003816) 119860
119895]
(9)
119899
prod
119895=1
1003816100381610038161003816100381610038161003816100381610038161003816
Γ119860119895(1 minus 119886
119895+
119903
sum
119896=1
120572(119896)
119895119904119896)
1003816100381610038161003816100381610038161003816100381610038161003816
le (2120587)sum119899
119895=1(1198601198952)
119899
prod
119895=1
(120572(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816)
[12minusR(119886119895)+120572(119896)
119895120590119896]119860119895
times exp[
[
minus
Π
2
119899
sum
119895=1
(120572(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816+
10038161003816100381610038161003816I (119886
119895)
10038161003816100381610038161003816) 119860
119895]
]
(10)
Similarly we have119901
prod
119895=119899+1
1003816100381610038161003816100381610038161003816100381610038161003816
Γ119860119895(119886
119895minus
119903
sum
119896=1
120572(119896)
119895119904119896)
1003816100381610038161003816100381610038161003816100381610038161003816
ge (2120587)sum119896
119895=119899+1(1198601198952)
119896
prod
119895=119899+1
(120572(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816)
[R(119886119895)minus120572(119896)
119895120590119896minus12]119860
119895
times exp[
[
minus
120587
2
119901
prod
119895=119899+1
(120572(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816+
10038161003816100381610038161003816I (119886
119895)
10038161003816100381610038161003816) 119860
119895]
]
119902
prod
119895=1
1003816100381610038161003816100381610038161003816100381610038161003816
Γ119861119895(1 minus 119887
119895+
119903
sum
119896=1
120573(119896)
119895119904119896)
1003816100381610038161003816100381610038161003816100381610038161003816
ge (2120587)sum119902
119895=1(1198611198952)
119902
prod
119895=1
(120573(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816)
[12minusR(119887119895)+120573(119896)
119895120590119896]119861119895
times exp[
[
minus
Π
2
119902
prod
119895=1
(120573(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816+
10038161003816100381610038161003816I (119886
119895)
10038161003816100381610038161003816) 119861
119895]
]
119899119896
prod
119895=1
100381610038161003816100381610038161003816
Γ119862(119896)
119895(1 minus 119888
(119896)
119895+ 120574
(119896)
119895119904119896)
100381610038161003816100381610038161003816
le (2120587)sum119899119896
119895=1(119862(119896)
1198952)
119899119896
prod
119895=1
(120574(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816)
[12minusR(119888(119896)
119895)+120574(119896)
119895120590119896]119862(119896)
119895
times exp[
[
minus
Π
2
119899119896
prod
119895=1
(120574(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816+
10038161003816100381610038161003816I (119888
(119896)
119895)
10038161003816100381610038161003816) 119862
(119896)
119895]
]
119901119896
prod
119895=119899119896+1
100381610038161003816100381610038161003816
Γ119862(119896)
119895(119888
(119896)
119895minus 120574
(119896)
119895119904119896)
100381610038161003816100381610038161003816
ge (2120587)sum119901119896
119895=119899119896+1(119862(119896)
1198952)
119901119896
prod
119895=119899119896+1
(120574(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816)
[R(119888(119896)
119895)minus120574(119896)
119895120590119896minus12]119862
(119896)
119895
times exp[
[
minus
120587
2
119901119896
prod
119895=119899119896+1
(120574(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816+
10038161003816100381610038161003816I (119888
(119896)
119895)
10038161003816100381610038161003816) 119862
(119896)
119895]
]
119898119896
prod
119895=1
100381610038161003816100381610038161003816
Γ119863(119896)
119895(119889
(119896)
119895minus 120575
(119896)
119895119904119896)
100381610038161003816100381610038161003816
le (2120587)sum119898119896
119895=1(119863(119896)
1198952)
119898119896
prod
119895=1
(120575(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816)
[R(119889(119896)
119895)+120575(119896)
119895120590119896minus12]119863
(119896)
119895
times exp[
[
minus
120587
2
119898119896
prod
119895=1
(120575(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816+
10038161003816100381610038161003816I (119889
(119896)
119895)
10038161003816100381610038161003816)119863
(119896)
119895]
]
119902119896
prod
119895=119898119896+1
100381610038161003816100381610038161003816
Γ119863(119896)
119895(1 minus 119889
(119896)
119895+ 120575
(119896)
119895119904119896)
100381610038161003816100381610038161003816
le (2120587)sum119902119896
119895=119898119896+1(119863(119896)
1198952)
119902119896
prod
119895=119898119896+1
(120575(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816)
[12minusR(119889(119896)
119895)+120575(119896)
119895120590119896]119863(119896)
119895
times exp[
[
minus
120587
2
119902119896
prod
119895=119898119896+1
(120575(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816+
10038161003816100381610038161003816I (119889
(119896)
119895)
10038161003816100381610038161003816)119863
(119896)
119895]
]
(11)
Also
119911119904119896
119896= exp [(120590
119896+ 119894119905
119896) (log 1003816
100381610038161003816119911119896
1003816100381610038161003816+ 119894 arg (119911
119896))]
= exp [120590119896log 1003816
100381610038161003816119911119896
1003816100381610038161003816minus 119905
119896arg (119911
119896)]
=1003816100381610038161003816119911119896
1003816100381610038161003816
120590119896 exp [minus119905
119896arg (119911
119896)]
(12)
Hence substituting (10)-(11) in (1) and using (12) we haveafter much simplification
1003816100381610038161003816120601 (119904
1 119904
119903) 120579
119896(119904119896) 119911
119904119896
119896
1003816100381610038161003816
sim 119862119896
1003816100381610038161003816119905119896
1003816100381610038161003816
Ω119896+120583119896120590119896 exp [minus119905
119896arg (119911
119896) minus
120587
2
1003816100381610038161003816119905119896
1003816100381610038161003816Δ119896]
(13)
where 119862119896is independent of 119905
119896and Δ
119896 120583
119896 and Ω
119896are given
by (6) (7) and (8) respectively for each 119896 = 1 2 119903Hence the result follows
4 International Journal of Engineering Mathematics
Remark 1 If 119863(119894)
119895= 1 (119895 = 1 119898
119894 119894 = 1 119903) in (1) then
the function will be denoted by
I[[
[
1199111
119911119903
]
]
]
= I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
times[
[
[
1199111
119911119903
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 1)
11198981
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)1198981+11199021
(119889(119903)
119895 120575
(119903)
119895 1)
1119898119903
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)119898119903+1119902119903
]
]
]
=
1
(2120587119894)119903int
L1
sdot sdot sdot int
L119903
120601 (1199041 119904
119903) 120579
1(1199041) sdot sdot sdot 120579
119903(119904119903) 119911
1199041
1sdot sdot sdot 119911
119904119903
119903119889119904
1sdot sdot sdot 119889119904
119903
(14)
where
120579119894(119904119894) =
119899119894
prod
119895=1
Γ119862(119894)
119895(1 minus 119888
(119894)
119895+ 120574
(119894)
119895119904119894)
119898119894
prod
119895=1
Γ (119889(119894)
119895minus 120575
(119894)
119895119904119894)
times(
119901119894
prod
119895=119899119894+1
Γ119862(119894)
119895(119888(119894)
119895minus 120574
(119894)
119895119904119894)
119902119894
prod
119895=119898119894+1
Γ119863(119894)
119895(1minus 119889
(119894)
119895+120575
(119894)
119895119904119894))
minus1
(15)where 119894 = 1 119903
Remark 2 If 119862(119894)
119895= 1 (119895 = 1 119899
119894) 119863(119894)
119895= 1 (119895 = 1 119898
119894)
where 119894 = 1 119903 and if 119899 = 0 in (1) then the correspondingfunction will be denoted by
I1
[
[
[
1199111
119911119903
]
]
]
= I001198981 119899111989811990311989911990311990111990211990111199021119901119903119902119903
times[
[
[
1199111
119911119903
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 1)
11198991
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)1198991+11199011
(119888(119903)
119895 120574
(119903)
119895 1)
1119899119903
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)119899119903+1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 1)
11198981
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)1198981+11199021
(119889(119903)
119895 120575
(119903)
119895 1)
1119898119903
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)119898119903+1119902119903
]
]
]
=
1
(2120587119894)119903int
L1
sdot sdot sdot int
L119903
1206011(1199041 119904
119903) 120579
1(1199041) sdot sdot sdot 120579
119903(119904119903) 119911
1199041
1sdot sdot sdot 119911
119904119903
119903119889119904
1sdot sdot sdot 119889119904
119903
(16)
where1206011(1199041 119904
119903)
=
1
prod119901
119895=1Γ119860119895(119886
119895minussum
119903
119894=1120572(119894)
119895119904119894) prod
119902
119895=1Γ119861119895(1minus119887
119895+sum
119903
119894=1120573(119894)
119895119904119894)
120579119894(119904119894) =
prod119899119894
119895=1Γ (1minus119888
(119894)
119895+120574
(119894)
119895119904119894)prod
119898119894
119895=1Γ (119889
(119894)
119895minus 120575
(119894)
119895119904119894)
prod119901119894
119895=119899119894+1
Γ119862(119894)
119895(119888
(119894)
119895minus120574
(119894)
119895119904119894)prod
119902119894
119895=119898119894+1
Γ119863(119894)
119895(1minus119889
(119894)
119895+120575
(119894)
119895119904119894)
forall119894 = 1 119903
(17)
4 Series Representationif
(i) 119911119894
= 0 (119894 = 1 119903) and 120583119894lt 0 where 120583
119894is given by
(4)
(ii) 120575(119894)
ℎ119894
(119889(119894)
119895+119896
119894) = 120575
(119894)
119895(119889
(119894)
ℎ119894
+120578119894) for 119895 = ℎ
119894 119895 ℎ
119894= 1 119898
119894
(119894 = 1 119903) 119896119894 120578
119894= 0 1 2 (119894 = 1 119903)
then
I[[
[
1199111
119911119903
]
]
]
=
1198981
sum
ℎ1=1
sdot sdot sdot
119898119903
sum
ℎ119903=1
infin
sum
1198961=1
sdot sdot sdot
infin
sum
119896119903=1
times [1206011(
119889ℎ(1)
1+ 119896
1
120575ℎ(1)
1
119889ℎ(119903)
119903+ 119896
119903
120575ℎ(119903)
119903
)
times
119903
prod
119894=1
(minus1)119896119894
120575ℎ(119894)
119894119896119894
119911(119889ℎ119894+119896119894)120575ℎ119894
119894]
119895 = ℎ119894
(18)
This result can be proved on computing the residues at the
International Journal of Engineering Mathematics 5
poles as follows
119904119903=
119889ℎ(119894)
119894+ 119896
119894
120575ℎ(119894)
119894
(ℎ119894= 1 119898
119894 119896
119894= 0 1 2 ) for 119894 = 1 119903
(19)
The behaviour of the function I[1199111119911119903
] is given by
I[[
[
1199111
119911119903
]
]
]
= 119874(
119903
prod
119895=1
1003816100381610038161003816119911119894
1003816100381610038161003816
120601119895
) max 10038161003816100381610038161199111
1003816100381610038161003816
1003816100381610038161003816119911119903
1003816100381610038161003816 997888rarr 0
(20)
where
120601119895= min
1le119895le119898119894
[
[
Re(119889(119894)
119895
120575(119894)
119895
)]
]
(119894 = 1 119903) (21)
On the other hand when |119911119894| rarr infin (119894 = 1 119903) the
associated function I1[
1199111119911119903
] given by (16) has the behaviour
I1
[
[
[
1199111
119911119903
]
]
]
= 119874(
119903
prod
119895=1
10038161003816100381610038161003816119911119895
10038161003816100381610038161003816
120601119895
) min 10038161003816100381610038161199111
1003816100381610038161003816
1003816100381610038161003816119911119903
1003816100381610038161003816 997888rarr 0
(22)
where
120601119895= max
1le119895le119899119894
[
[
Re(1 minus 119888
(119894)
119895
120574(119894)
119895
)]
]
(119894 = 1 119903) (23)
5 Elementary Special Cases
In this section we mention some interesting and usefulspecial cases of the I-function of ldquo119903rdquo variables
(i) If all the exponents 119860119895(119895 = 1 119901) 119861
119895(119895 =
1 119902) 119862(119894)
119895(119895 = 1 119901
119894 119894 = 1 119903) and
119863(119894)
119895(119895 = 1 119902
119894 119894 = 1 119903) in (1) are equal to
unity we obtain H-function of ldquo119903rdquo variables definedby Srivastava and Panda [7]
(ii) When 119901 = 119902 = 119899 = 0 (1) degenerates into the productof 119903mutually independent I- functions of one variableintroduced by Rathie [1]
(iii) When 119901 = 119902 = 119899 = 0 and 119903 = 1 (1) reduces to theI-function defined by Rathie [1]
(iv) When 119899 = 119901 119898119894= 1 119899
119894= 119901
119894 119894 = 1 119903 and 119860
119895=
119861119895= 119862
119895= 119863
119895= 1 and (119889
(119894)
119895 120575
(119894)
119895 119863
(119894)
119895) is replaced by
(0 1 1) (119889(119894)119895 120575
(119894)
119895 119863
(119894)
119895) (1) reduces to the generalized
Lauricella function [10]
I011990111199011111990111990311990111990211990111199021+1119901
119903119902119903+1
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 1)
1119901 (119888
(1)
119895 120574
(1)
119895 1)
11199011
(119888(119903)
119895 120574
(119903)
119895 1)
1119901119903
(119887119895 120573
(1)
119895 120573
(1)
119895 120573
(119903)
119895 1)
1119902 (0 1 1) (119889
(1)
119895 120575
(1)
119895 1)
11199021
(0 1 1) (119889(119903)
119895 120575
(119903)
119895 1)
1119902119903
]
]
]
=
prod119901
119895=1Γ (1 minus 119886
119895)prod
1199011
119895=1Γ (1 minus 119888
(1)
119895) sdot sdot sdotprod
119901119903
119895=1Γ (1 minus 119888
(119903)
119895)
prod119902
119895=1Γ (1 minus 119887
119895)prod
1199021
119895=1Γ (1 minus 119889
(1)
119895) sdot sdot sdotprod
119902119903
119895=1Γ (1 minus 119889
(119903)
119895)
times 1198651199011199011119901119903
1199021199021119902119903
[
[
[
minus1199111
minus119911
119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(1 minus 119886119895 120572
(1)
119895 120572
(119903)
119895)1119901
(1 minus 119888(1)
119895 120574
(1)
119895)11199011
(1 minus 119888((119903))
119895 120574
(119903)
119895)1119901119903
(1 minus 119887119895 120573
(1)
119895 120573
(119903)
119895)1119902
(1 minus 119889(1)
119895 120575
(1)
119895)11199021
(1 minus 119889(119903)
119895 120575
(119903)
119895)1119902119903
]
]
]
(24)
(v) I001119901111199011199030011990111199021+1119901
119903119902119903+1
[
[
[
minus1199111
minus119911
119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
mdash (1 minus 119888(1)
119895 120574
(1)
119895 1)
11199011
(1 minus 119888(119903)
119895 120574
(119903)
119895 1)
1119901119903
mdash (0 1 1) (1 minus 119889(1)
119895 120575
(1)
119895 1)
11199021
(0 1 1) (1 minus 119889(119903)
119895 120575
(119903)
119895 1)
1119902119903
]
]
]
=1199011
Ψ1199021
[
[
(119888(1)
119895 120574
(1)
119895)11199011
(119889(1)
119895 120575
(1)
119895)11199021
1199111]
]
times sdot sdot sdot times119901119903
Ψ119902119903
[
[
(119888(119903)
119895 120574
(119903)
119895)1119901119903
(119889(119903)
119895 120575
(119903)
119895)1119902119903
119911119903]
]
(25)
6 International Journal of Engineering Mathematics
where the functions119901119894
Ψ119902119894
119894 = 1 119903 are Wrightrsquos general-ized hypergeometric functions [11]
(vi) I001010000202
[
[
[
1199111
119911119903
10038161003816100381610038161003816100381610038161003816
mdash mdash mdashmdash (0 1 1) (minus120583
1 120572
1 1) (0 1 1) (minus120583
119903 120572
119903 1)
]
]
]
=
119903
prod
119894=1
119869120572119894
120583119894
(119911119894) (26)
where the functions 119869120572119894
120583119894
(119911119894) are Wrightrsquos generalized Bessel
functions [12]
(vii) I001212002222
[
[
[
minus1199111
minus119911
119903
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
mdash (1 1 1) (1-1205721 1 120583
1) (1 1 1) (1-120572
119903 1 120583
119903)
mdash (0 1 1) (-1205721 1 120583
1) (0 1 1) (-120572
119903 1 120583
119903)
]
]
]
=
119903
prod
119894=1
Φ(119911119894 120583
119894 120572
119894) (27)
where Φ(119911119894 120583
119894 120572
119894) 119894 = 1 119903 are the generalized Riemann
zeta functions [13 page 27 111 (1)] which are the generaliza-tions of Hurwitz zeta functions and Riemann zeta functions[13 page 24 110 (1) and 112 (1)]
(viii) I001212002222
[
[
[
minus1199111
minus119911
119903
10038161003816100381610038161003816100381610038161003816100381610038161003816
mdash (1 1 1) (1 1 1205831) (1 1 1) (1 1 120583
119903)
mdash (0 1 1) (0 1 1205831) (0 1 1) (0 1 120583
119903)
]
]
]
=
119903
prod
119894=1
119865 (119911119894 120583
119894) (28)
where 119865(119911119894 120583
119894) are the polylogarithms of order 120583
119894 For 120583
119894= 2
119894 = 1 119903 theRHS of (28) reduces to the product of Eulerrsquosdilogarithm [13 page 31 1111 equation (2)]
6 Elementary Properties andTransformation Formulas
Theproperties given below are immediate consequence of thedefinition (1) and hence they are given here without proof
(i) I001198981 119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895119861119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I001198991 1198981119899119903 11989811990311990211990111990211199011119902119903119901119903
[
[
[
119911minus1
1
119911minus1
119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(1 minus 119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(1 minus 119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(1 minus 119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
(1 minus 119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(1 minus 119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(1 minus 119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
]
]
]
(29)
(ii) 1199111198961
1sdot sdot sdot 119911
119896119903
119903I [119911
1sdot sdot sdot 119911
119903]
= I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
[
[
1199111
119911119903
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895+
119903
sum
119894=1
119896119894120572(119894)
119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)
1119901
(119888(1)
119895+119896
1120574(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895+119896
119903120574(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895+
119903
sum
119894=1
119896119894120573(119894)
119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)
1119902
(119889(1)
119895+ 119896
1120575(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895+ 119896
119903120575(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
]
]
(30)
for 119896119894gt 0 119894 = 1 119903
International Journal of Engineering Mathematics 7
(iii) 1
1198961
sdot sdot sdot
1
119896119903
I [1199111 119911
119903]
= I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
[
1199111198961
1
119911119896119903
119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 119896
1120572(1)
119895 119896
119903120572(119903)
119895 119860
119895)1119901
(119888(1)
119895 119896
1120574(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 119896
119903120574(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 119896
1120573(1)
119895 119896
119903120573(119903)
119895 119861
119895)1119901
(119889(1)
119895 119896
1120575(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 119896
119903120575(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
]
(31)
where 119896119894gt 0 119894 = 1 119903
(iv) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886 120572 0 0 119860) (119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I0119899minus111989811198991+1119898119903 1198991199031199011199021199011+11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 )
2119901 (119886 120572 119860) (119888
(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(32)
where 119901 ge 119899 ge 1
(v) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119886 120572 0 0 119860) (119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I011989911989811198991 119898119903 119899119903119901minus1119902119901
1+11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119886 120572 119860) (119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(33)
where 119901 minus 1 ge 119899 ge 0
(vi) I011989911989811198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119887 120573 0 0 119861) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I011989911989811198991 119898119903 119899119903119901119902minus1119901
11199021+1119901
119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119887 120573 119861) (119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(34)
where 119902 minus 1 ge 0
8 International Journal of Engineering Mathematics
(vii) I01198991198981 1198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886 0 0 119860) (119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= Γ119860(1 minus 119886) times I0119899minus11198981 1198991119898119903119899119903
119901minus111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(35)
where 119901 ge 119899 ge 1R(1 minus 119886) gt 0
(viii) I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119886 0 0 119860) (119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119860(119886)
times I011989911989811198991119898119903119899119903119901minus1119902119901
11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(36)
where 119901 minus 1 ge 119899 ge 0R(119886) gt 0
(ix) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119887 0 0 119861) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119861(1 minus 119887)
times I011989911989811198991 119898119903 119899119903119901119902minus1119901
11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(37)
where 119902 minus 1 ge 0R(1 minus 119887) gt 0
(x) I011989911989811198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888 0 119862) (119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= Γ119862(1 minus 119888) times I011989911989811198991minus1119898119903 119899119903
1199011199021199011minus11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(38)
where 1199011ge 119899
1ge 1R(1 minus 119888) gt 0
International Journal of Engineering Mathematics 9
(xi) I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888 0 119862) (119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119862(119888)
times I011989911989811198991 119898119903 1198991199031199011199021199011minus11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(39)
where 1199011minus 1 ge 119899
1ge 0R(119888) gt 0
(xii) I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889 0 119863) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= Γ119863(119889) times I01198991198981minus11198991 119898119903119899119903
11990111990211990111199021minus1119901
119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(40)
where 1199021ge 119898
1ge 1R(119889) gt 0
(xiii) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889 0 119863) (119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119863(1 minus 119889)
times I01198991198981119899111989811990311989911990311990111990211990111199021minus1119901
119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(41)
provided that 1199021minus 1 ge 119898
1ge 0R(1 minus 119889) ge 0
(xiv) I01198991198981 1198991 119898119903 1198991199031199011199021199011 1199021 119901119903 119902119903
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902minus1
(1198861 120572(1)
1 120572
(119903)
1 1198601) (119889
1
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119888(1)
1 120574
(1)
1 119862
(1)
1) (119889
(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
(119888(119903)
1 120574
(119903)
1 119862
(119903)
1)
]]]
]
= I0119899minus11198981 1198991minus1119898119903 119899119903minus1119901minus1119902minus11199011minus11199021minus1119901119903minus1119902119903minus1
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902minus1
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
]]]
]
(42)
provided that 119901 ge 119899 ge 1 119901119894ge 119899
119894ge 1 119894 = 1 119903 and 119902 ge 1
119902119894ge 119898
119894+ 1 119894 = 1 119903
10 International Journal of Engineering Mathematics
(xv) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119889(1)
1 120575
(1)
1 119863
(1)
1) (119888
(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119889(119903)
1 120575
(119903)
1 119863
(119903)
1)
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I01198991198981minus11198991 119898119903minus11198991199031199011199021199011minus11199021minus1119901
119903minus1119902119903minus1
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
(43)
provided that 119901 ge 119899 + 1 119902 ge 1 119901119894ge 119899
119894+ 1 and 119902
119894ge
119898119894ge 1 119894 = 1 119903
(xvi) I011989911989811198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119888(1)
1199011
120574(1)
1199011
119862(1)
1199011
) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119888(119903)
119901119903
120574(119903)
119901119903
119862(119903)
119901119903
) (119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
= I0119899 1198981minus11198991 119898
119903minus1119899119903
119901119902 1199011minus11199021minus1 119901
119903minus1119902119903minus1
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
(44)
provided that 119901119894ge 119899
119894+ 1 119902
119894ge 119898
119894 119894 = 1 119903
(xvii) I01198991198981 1198991 119898119903 1198991199031199011199021199011 1199021 119901119903 119902119903
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119889(1)
1199021 120575
(1)
1199021 119863
(1)
1199021) (119888
(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119889(119903)
119902119903 120575
(119903)
119902119903 119863
(119903)
119902119903) (119888
(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]]]
]
= I01198991198981 1198991minus1119898119903 119899119903minus11199011199021199011minus11199021minus1119901119903minus1119902119903minus1
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
]]]
]
(45)
where 119901 ge 119899 119902 ge 1 119901119894ge 119899
119894ge 1 119902
119894minus 1 ge 119898
119894 119894 = 1 119903
7 Special Cases
When 119903 = 2 and all the exponents 119860119895(119895 = 1 119901)
119861119895(119895 = 1 119902) 119862
(119894)
119895(119895 = 1 119901
119894 119894 = 1 119903) and
119863(119894)
119895(1 119902
119894 119894 = 1 119903) the I-function of ldquo119903rdquo variables
reduces to H-function of two variables and therefore weobtain the corresponding results in H-function of two vari-ables [14]
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgment
The authors are immensely grateful to the worthy referee forsome useful and valuable suggestions for the improvement ofthis paper which led to a better presentation
International Journal of Engineering Mathematics 11
References
[1] A K Rathie ldquoA new generalization of generalized hypergeo-metric functionsrdquo Le Matematiche vol 52 no 2 pp 297ndash3101997
[2] I S Ansari F Yilmaz M S Alouni and O Kucur ldquoNewresults on the sum of Gamma random variates with applicationto the performance of wireless communication systems overnakagami-m fading channelsrdquo httparxiv-web3librarycornelleduabs12022576v4
[3] I S Ansari andYilmaz ldquoOn the sumof squared n-Randomvari-ates with application to the performance of wireless communi-cation systemsrdquo httparxiv-web3librarycornelleduabs12100100v1
[4] X Minghua W Yik-Chung and A Sonia ldquoExact outage prob-ability of dual-hop CSI-assisted AF relaying over nakagami-mfading channelsrdquo IEEE Transactions on Signal Processing vol60 no 10 pp 5578ndash5583 2012
[5] K ShanthaKumari T M Vasudevan Nambisan and AK Rathie ldquoA study of the I-function of two variablesrdquohttparxivorgabs12126717
[6] P K Mittal and K C Gupta ldquoAn integral involving generalizedfunction of two variablesrdquo Proceedings of the Indian Academy ofSciences A vol 75 no 3 pp 117ndash123 1972
[7] HM Srivastava and R Panda ldquoSome bilateral generating func-tions for a class of generalkized hypergeometric polynomialsrdquoJournal Fur die Reine und Angewandte Mathematik vol 17 no288 pp 265ndash274 1976
[8] B L J Braaksma ldquoAsymptotic expansions and analytic continu-ations for a class of Barnesintegralsrdquo Compositio Mathematicahvol 15 pp 239ndash341 1964
[9] Y L LukeThe Special Functions andTheir Approximations vol1 Academic Press New York NY USA 1969
[10] H M Srivastava and M C Daoust ldquoOn Eulerian integralsassociated with Kampe de Ferietrsquos functionrdquo Publications DeLrsquoInstitut Mathematique vol 9 no 23 pp 199ndash202 1969
[11] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Halsted Press Chichester UK
[12] E M Wright ldquoThe asymptotic expansion of the generalizedBessel functionrdquo Proceedings of London Mathematical Societyvol 38 pp 257ndash270 1935
[13] A Erdelyi Higher Transcendental Functions vol 1 McGraw-Hill New York NY USA 1953
[14] H M Srivastava K C Gupta and S P Goyal The HmdashFunctions of One and Two Variables With Applications SouthAsian Publishers New Delhi India 1982
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 International Journal of Engineering Mathematics
where 120601(1199041 119904
119903) 120579
119894(119904119894) 119894 = 1 119903 are given by
120601 (1199041 119904
119903) =
119899
prod
119895=1
Γ119860119895(1 minus 119886
119895+
119903
sum
119894=1
120572(119894)
119895119904119894)
times (
119901
prod
119895=119899+1
Γ119860119895(119886
119895minus
119903
sum
119894=1
120572(119894)
119895119904119894)
times
119902
prod
119895=1
Γ119861119895(1 minus 119887
119895+
119903
sum
119894=1
120573(119894)
119895119904119894))
minus1
(2)
120579119894(119904119894) = (
119899119894
prod
119895=1
Γ119862(119894)
119895(1 minus 119888
(119894)
119895+ 120574
(119894)
119895119904119894)
times
119898119894
prod
119895=1
Γ119863(119894)
119895(119889
(119894)
119895minus 120575
(119894)
119895119904119894))
times (
119901119894
prod
119895=119899119894+1
Γ119862(119894)
119895(119888
(119894)
119895minus 120574
(119894)
119895119904119894)
times
119902119894
prod
119895=119898119894+1
Γ119863(119894)
119895(1 minus 119889
(119894)
119895+ 120575
(119894)
119895119904119894))
minus1
(3)
where 119894 = 1 119903Also
(i) 119911119894
= 0 for 119894 = 1 119903(ii) 119894 = radicminus1(iii) an empty product is interpreted as unity(iv) the parameters 119898
119895 119899
119895 119901
119895 119902
119895(119895 = 1 119903) 119899 119901 and
119902 are nonnegative integers such that 0 le 119899 le 119901 119902 ge 00 le 119899
119895le 119901
119895 and 0 le 119898
119895le 119902
119895(119895 = 1 119903) (not all
zero simulataneously)(v) 120572
(119894)
119895(119895 = 1 119901 119894 = 1 119903) 120573(119894)
119895(119895 = 1 119902 119894 =
1 119903) 120574(119894)119895
(119895 = 1 119901119894 119894 = 1 119903) and 120575
(119894)
119895(119895 =
1 119902119894 119894 = 1 119903) are assumed to be positive
quantities for standardisation purpose However thedefinition of I-function of ldquo119903rdquo variables will have ameaning even if some of the quantities are zero ornegative numbers For these we may obtain cor-responding transformation formulas which will begiven in a later section
(vi) 119886119895(119895 = 1 119901) 119887
119895(119895 = 1 119902) 119888(119894)
119895(119895 = 1 119901
119894
119894 = 1 119903) and 119889(119894)
119895(119895 = 1 119902
119894 119894 = 1 119903) are
complex numbers(vii) the exponents 119860
119895(119895 = 1 119901) 119861
119895(119895 = 1 119902)
119862(119894)
119895(119895 = 1 119901
119894 119894 = 1 119903) and 119863
(119894)
119895(119895 =
1 119902119894 119894 = 1 119903) of various gamma functions
involved in (2) and (3) may take noninteger values(viii) the contour L
119894in the complex 119904
119894-plane is of Mellin-
Barnes type which runs from 119888minus 119894infin to 119888+ 119894infin (119888 real)with indentation if necessary in such a manner thatall singularities of Γ119863
(119894)
119895(119889
(119894)
119895minus 120575
(119894)
119895119904119894) 119895 = 1 119898
119894lie
to the right and Γ119862(119894)
119895(1 minus 119888
(119894)
119895+ 120574
(119894)
119895119904119894) 119895 = 1 119899
119894are
to the left ofL119894
Following the results of Braaksma [8] the I-function of ldquo119903rdquovariables is analytic if
120583119894=
119901
sum
119895=1
119860119895120572(119894)
119895minus
119902
sum
119895=1
119861119895120573(119894)
119895+
119901119894
sum
119895=1
119862(119894)
119895120574(119894)
119895
minus
119902119894
sum
119895=1
119863(119894)
119895120575(119894)
119895le 0 119894 = 1 119903
(4)
3 Convergence Conditions
Integral (1) converges absolutely if
1003816100381610038161003816arg (119911
119896)1003816100381610038161003816lt
1
2
Δ119896120587 119896 = 1 119903 (5)
where
Δ119896=
[
[
minus
119901
sum
119895=119899+1
119860119895120572(119896)
119895minus
119902
sum
119895=1
119861119895120573(119896)
119895
+
119898119896
sum
1
119863(119896)
119895120575(119896)
119895minus
119902119896
sum
119898119896+1
119863(119896)
119895120575(119896)
119895
+
119899119896
sum
119895=1
119862(119896)
119895120574(119896)
119895minus
119901119896
sum
119895=119899119896+1
119862(119896)
119895120574(119896)
119895]
]
gt 0
(6)
and if | arg(119911119896)| = (12)Δ
119896120587 and Δ
119896ge 0 119896 = 1 119903
then integral (1) converges absolutely under the followingconditions
(i) 120583119896= 0 Ω
119896lt minus1 where 120583
119896is given by (4) and
Ω119896=
119901
sum
119895=1
[
1
2
minus R (119886119895)]119860
119895
minus
119902
sum
119895=1
[
1
2
minus R (119887119895)] 119861
119895
+
119901119896
sum
119895=1
[
1
2
minus R (119888(119896)
119895)]119862
(119896)
119895
minus
119902119896
sum
119895=1
[
1
2
minus R (119889(119896)
119895)]119863
(119896)
119895
119896 = 1 119903
(7)
(ii) 120583119896
= 0 with 119904119896
= 120590119896+ 119894119905
119896 (120590
119896and 119905
119896are real 119896 =
1 119903) and 120590119896are chosen so that for |119905
119896| rarr infin we
have Ω119896+ 120590
119896120583119896lt minus1
Outline of the ProofThe convergence of integral (1) dependson the asymptotic behaviour of the functions 120601(119904
1 119904
119903)
120579119894(119904119894) 119894 = 1 119903 defined by (2) and (3) respectively Such
International Journal of Engineering Mathematics 3
asymptotic behaviour is based on the following relation forthe gamma function Γ(119911) 119911 = 119909 + 119894119910 119909 119910 isin R [9]
1003816100381610038161003816Γ (119909 + 119894119910)
1003816100381610038161003816sim radic2120587
10038161003816100381610038161199101003816100381610038161003816
119909minus12 exp(minus
1
2
12058710038161003816100381610038161199101003816100381610038161003816)
10038161003816100381610038161199101003816100381610038161003816rarr infin
(8)
Along the contour L119896 if we put 119904
119896= 120590
119896+ 119894119905
119896and take the
limit as |119905119896| rarr infin for 119896 = 1 119903 we obtain by virtue of (8)
that1003816100381610038161003816100381610038161003816100381610038161003816
Γ119860119895(1 minus 119886
119895+
119903
sum
119896=1
120572(119896)
119895119904119896)
1003816100381610038161003816100381610038161003816100381610038161003816
le (2120587)1198601198952(120572
(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816)
[12minusR(119886119895)+120572(119896)
119895120590119896]119860119895
times exp [minus
120587
2
(120572(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816+
10038161003816100381610038161003816I (119886
119895)
10038161003816100381610038161003816) 119860
119895]
(9)
119899
prod
119895=1
1003816100381610038161003816100381610038161003816100381610038161003816
Γ119860119895(1 minus 119886
119895+
119903
sum
119896=1
120572(119896)
119895119904119896)
1003816100381610038161003816100381610038161003816100381610038161003816
le (2120587)sum119899
119895=1(1198601198952)
119899
prod
119895=1
(120572(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816)
[12minusR(119886119895)+120572(119896)
119895120590119896]119860119895
times exp[
[
minus
Π
2
119899
sum
119895=1
(120572(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816+
10038161003816100381610038161003816I (119886
119895)
10038161003816100381610038161003816) 119860
119895]
]
(10)
Similarly we have119901
prod
119895=119899+1
1003816100381610038161003816100381610038161003816100381610038161003816
Γ119860119895(119886
119895minus
119903
sum
119896=1
120572(119896)
119895119904119896)
1003816100381610038161003816100381610038161003816100381610038161003816
ge (2120587)sum119896
119895=119899+1(1198601198952)
119896
prod
119895=119899+1
(120572(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816)
[R(119886119895)minus120572(119896)
119895120590119896minus12]119860
119895
times exp[
[
minus
120587
2
119901
prod
119895=119899+1
(120572(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816+
10038161003816100381610038161003816I (119886
119895)
10038161003816100381610038161003816) 119860
119895]
]
119902
prod
119895=1
1003816100381610038161003816100381610038161003816100381610038161003816
Γ119861119895(1 minus 119887
119895+
119903
sum
119896=1
120573(119896)
119895119904119896)
1003816100381610038161003816100381610038161003816100381610038161003816
ge (2120587)sum119902
119895=1(1198611198952)
119902
prod
119895=1
(120573(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816)
[12minusR(119887119895)+120573(119896)
119895120590119896]119861119895
times exp[
[
minus
Π
2
119902
prod
119895=1
(120573(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816+
10038161003816100381610038161003816I (119886
119895)
10038161003816100381610038161003816) 119861
119895]
]
119899119896
prod
119895=1
100381610038161003816100381610038161003816
Γ119862(119896)
119895(1 minus 119888
(119896)
119895+ 120574
(119896)
119895119904119896)
100381610038161003816100381610038161003816
le (2120587)sum119899119896
119895=1(119862(119896)
1198952)
119899119896
prod
119895=1
(120574(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816)
[12minusR(119888(119896)
119895)+120574(119896)
119895120590119896]119862(119896)
119895
times exp[
[
minus
Π
2
119899119896
prod
119895=1
(120574(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816+
10038161003816100381610038161003816I (119888
(119896)
119895)
10038161003816100381610038161003816) 119862
(119896)
119895]
]
119901119896
prod
119895=119899119896+1
100381610038161003816100381610038161003816
Γ119862(119896)
119895(119888
(119896)
119895minus 120574
(119896)
119895119904119896)
100381610038161003816100381610038161003816
ge (2120587)sum119901119896
119895=119899119896+1(119862(119896)
1198952)
119901119896
prod
119895=119899119896+1
(120574(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816)
[R(119888(119896)
119895)minus120574(119896)
119895120590119896minus12]119862
(119896)
119895
times exp[
[
minus
120587
2
119901119896
prod
119895=119899119896+1
(120574(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816+
10038161003816100381610038161003816I (119888
(119896)
119895)
10038161003816100381610038161003816) 119862
(119896)
119895]
]
119898119896
prod
119895=1
100381610038161003816100381610038161003816
Γ119863(119896)
119895(119889
(119896)
119895minus 120575
(119896)
119895119904119896)
100381610038161003816100381610038161003816
le (2120587)sum119898119896
119895=1(119863(119896)
1198952)
119898119896
prod
119895=1
(120575(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816)
[R(119889(119896)
119895)+120575(119896)
119895120590119896minus12]119863
(119896)
119895
times exp[
[
minus
120587
2
119898119896
prod
119895=1
(120575(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816+
10038161003816100381610038161003816I (119889
(119896)
119895)
10038161003816100381610038161003816)119863
(119896)
119895]
]
119902119896
prod
119895=119898119896+1
100381610038161003816100381610038161003816
Γ119863(119896)
119895(1 minus 119889
(119896)
119895+ 120575
(119896)
119895119904119896)
100381610038161003816100381610038161003816
le (2120587)sum119902119896
119895=119898119896+1(119863(119896)
1198952)
119902119896
prod
119895=119898119896+1
(120575(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816)
[12minusR(119889(119896)
119895)+120575(119896)
119895120590119896]119863(119896)
119895
times exp[
[
minus
120587
2
119902119896
prod
119895=119898119896+1
(120575(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816+
10038161003816100381610038161003816I (119889
(119896)
119895)
10038161003816100381610038161003816)119863
(119896)
119895]
]
(11)
Also
119911119904119896
119896= exp [(120590
119896+ 119894119905
119896) (log 1003816
100381610038161003816119911119896
1003816100381610038161003816+ 119894 arg (119911
119896))]
= exp [120590119896log 1003816
100381610038161003816119911119896
1003816100381610038161003816minus 119905
119896arg (119911
119896)]
=1003816100381610038161003816119911119896
1003816100381610038161003816
120590119896 exp [minus119905
119896arg (119911
119896)]
(12)
Hence substituting (10)-(11) in (1) and using (12) we haveafter much simplification
1003816100381610038161003816120601 (119904
1 119904
119903) 120579
119896(119904119896) 119911
119904119896
119896
1003816100381610038161003816
sim 119862119896
1003816100381610038161003816119905119896
1003816100381610038161003816
Ω119896+120583119896120590119896 exp [minus119905
119896arg (119911
119896) minus
120587
2
1003816100381610038161003816119905119896
1003816100381610038161003816Δ119896]
(13)
where 119862119896is independent of 119905
119896and Δ
119896 120583
119896 and Ω
119896are given
by (6) (7) and (8) respectively for each 119896 = 1 2 119903Hence the result follows
4 International Journal of Engineering Mathematics
Remark 1 If 119863(119894)
119895= 1 (119895 = 1 119898
119894 119894 = 1 119903) in (1) then
the function will be denoted by
I[[
[
1199111
119911119903
]
]
]
= I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
times[
[
[
1199111
119911119903
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 1)
11198981
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)1198981+11199021
(119889(119903)
119895 120575
(119903)
119895 1)
1119898119903
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)119898119903+1119902119903
]
]
]
=
1
(2120587119894)119903int
L1
sdot sdot sdot int
L119903
120601 (1199041 119904
119903) 120579
1(1199041) sdot sdot sdot 120579
119903(119904119903) 119911
1199041
1sdot sdot sdot 119911
119904119903
119903119889119904
1sdot sdot sdot 119889119904
119903
(14)
where
120579119894(119904119894) =
119899119894
prod
119895=1
Γ119862(119894)
119895(1 minus 119888
(119894)
119895+ 120574
(119894)
119895119904119894)
119898119894
prod
119895=1
Γ (119889(119894)
119895minus 120575
(119894)
119895119904119894)
times(
119901119894
prod
119895=119899119894+1
Γ119862(119894)
119895(119888(119894)
119895minus 120574
(119894)
119895119904119894)
119902119894
prod
119895=119898119894+1
Γ119863(119894)
119895(1minus 119889
(119894)
119895+120575
(119894)
119895119904119894))
minus1
(15)where 119894 = 1 119903
Remark 2 If 119862(119894)
119895= 1 (119895 = 1 119899
119894) 119863(119894)
119895= 1 (119895 = 1 119898
119894)
where 119894 = 1 119903 and if 119899 = 0 in (1) then the correspondingfunction will be denoted by
I1
[
[
[
1199111
119911119903
]
]
]
= I001198981 119899111989811990311989911990311990111990211990111199021119901119903119902119903
times[
[
[
1199111
119911119903
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 1)
11198991
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)1198991+11199011
(119888(119903)
119895 120574
(119903)
119895 1)
1119899119903
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)119899119903+1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 1)
11198981
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)1198981+11199021
(119889(119903)
119895 120575
(119903)
119895 1)
1119898119903
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)119898119903+1119902119903
]
]
]
=
1
(2120587119894)119903int
L1
sdot sdot sdot int
L119903
1206011(1199041 119904
119903) 120579
1(1199041) sdot sdot sdot 120579
119903(119904119903) 119911
1199041
1sdot sdot sdot 119911
119904119903
119903119889119904
1sdot sdot sdot 119889119904
119903
(16)
where1206011(1199041 119904
119903)
=
1
prod119901
119895=1Γ119860119895(119886
119895minussum
119903
119894=1120572(119894)
119895119904119894) prod
119902
119895=1Γ119861119895(1minus119887
119895+sum
119903
119894=1120573(119894)
119895119904119894)
120579119894(119904119894) =
prod119899119894
119895=1Γ (1minus119888
(119894)
119895+120574
(119894)
119895119904119894)prod
119898119894
119895=1Γ (119889
(119894)
119895minus 120575
(119894)
119895119904119894)
prod119901119894
119895=119899119894+1
Γ119862(119894)
119895(119888
(119894)
119895minus120574
(119894)
119895119904119894)prod
119902119894
119895=119898119894+1
Γ119863(119894)
119895(1minus119889
(119894)
119895+120575
(119894)
119895119904119894)
forall119894 = 1 119903
(17)
4 Series Representationif
(i) 119911119894
= 0 (119894 = 1 119903) and 120583119894lt 0 where 120583
119894is given by
(4)
(ii) 120575(119894)
ℎ119894
(119889(119894)
119895+119896
119894) = 120575
(119894)
119895(119889
(119894)
ℎ119894
+120578119894) for 119895 = ℎ
119894 119895 ℎ
119894= 1 119898
119894
(119894 = 1 119903) 119896119894 120578
119894= 0 1 2 (119894 = 1 119903)
then
I[[
[
1199111
119911119903
]
]
]
=
1198981
sum
ℎ1=1
sdot sdot sdot
119898119903
sum
ℎ119903=1
infin
sum
1198961=1
sdot sdot sdot
infin
sum
119896119903=1
times [1206011(
119889ℎ(1)
1+ 119896
1
120575ℎ(1)
1
119889ℎ(119903)
119903+ 119896
119903
120575ℎ(119903)
119903
)
times
119903
prod
119894=1
(minus1)119896119894
120575ℎ(119894)
119894119896119894
119911(119889ℎ119894+119896119894)120575ℎ119894
119894]
119895 = ℎ119894
(18)
This result can be proved on computing the residues at the
International Journal of Engineering Mathematics 5
poles as follows
119904119903=
119889ℎ(119894)
119894+ 119896
119894
120575ℎ(119894)
119894
(ℎ119894= 1 119898
119894 119896
119894= 0 1 2 ) for 119894 = 1 119903
(19)
The behaviour of the function I[1199111119911119903
] is given by
I[[
[
1199111
119911119903
]
]
]
= 119874(
119903
prod
119895=1
1003816100381610038161003816119911119894
1003816100381610038161003816
120601119895
) max 10038161003816100381610038161199111
1003816100381610038161003816
1003816100381610038161003816119911119903
1003816100381610038161003816 997888rarr 0
(20)
where
120601119895= min
1le119895le119898119894
[
[
Re(119889(119894)
119895
120575(119894)
119895
)]
]
(119894 = 1 119903) (21)
On the other hand when |119911119894| rarr infin (119894 = 1 119903) the
associated function I1[
1199111119911119903
] given by (16) has the behaviour
I1
[
[
[
1199111
119911119903
]
]
]
= 119874(
119903
prod
119895=1
10038161003816100381610038161003816119911119895
10038161003816100381610038161003816
120601119895
) min 10038161003816100381610038161199111
1003816100381610038161003816
1003816100381610038161003816119911119903
1003816100381610038161003816 997888rarr 0
(22)
where
120601119895= max
1le119895le119899119894
[
[
Re(1 minus 119888
(119894)
119895
120574(119894)
119895
)]
]
(119894 = 1 119903) (23)
5 Elementary Special Cases
In this section we mention some interesting and usefulspecial cases of the I-function of ldquo119903rdquo variables
(i) If all the exponents 119860119895(119895 = 1 119901) 119861
119895(119895 =
1 119902) 119862(119894)
119895(119895 = 1 119901
119894 119894 = 1 119903) and
119863(119894)
119895(119895 = 1 119902
119894 119894 = 1 119903) in (1) are equal to
unity we obtain H-function of ldquo119903rdquo variables definedby Srivastava and Panda [7]
(ii) When 119901 = 119902 = 119899 = 0 (1) degenerates into the productof 119903mutually independent I- functions of one variableintroduced by Rathie [1]
(iii) When 119901 = 119902 = 119899 = 0 and 119903 = 1 (1) reduces to theI-function defined by Rathie [1]
(iv) When 119899 = 119901 119898119894= 1 119899
119894= 119901
119894 119894 = 1 119903 and 119860
119895=
119861119895= 119862
119895= 119863
119895= 1 and (119889
(119894)
119895 120575
(119894)
119895 119863
(119894)
119895) is replaced by
(0 1 1) (119889(119894)119895 120575
(119894)
119895 119863
(119894)
119895) (1) reduces to the generalized
Lauricella function [10]
I011990111199011111990111990311990111990211990111199021+1119901
119903119902119903+1
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 1)
1119901 (119888
(1)
119895 120574
(1)
119895 1)
11199011
(119888(119903)
119895 120574
(119903)
119895 1)
1119901119903
(119887119895 120573
(1)
119895 120573
(1)
119895 120573
(119903)
119895 1)
1119902 (0 1 1) (119889
(1)
119895 120575
(1)
119895 1)
11199021
(0 1 1) (119889(119903)
119895 120575
(119903)
119895 1)
1119902119903
]
]
]
=
prod119901
119895=1Γ (1 minus 119886
119895)prod
1199011
119895=1Γ (1 minus 119888
(1)
119895) sdot sdot sdotprod
119901119903
119895=1Γ (1 minus 119888
(119903)
119895)
prod119902
119895=1Γ (1 minus 119887
119895)prod
1199021
119895=1Γ (1 minus 119889
(1)
119895) sdot sdot sdotprod
119902119903
119895=1Γ (1 minus 119889
(119903)
119895)
times 1198651199011199011119901119903
1199021199021119902119903
[
[
[
minus1199111
minus119911
119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(1 minus 119886119895 120572
(1)
119895 120572
(119903)
119895)1119901
(1 minus 119888(1)
119895 120574
(1)
119895)11199011
(1 minus 119888((119903))
119895 120574
(119903)
119895)1119901119903
(1 minus 119887119895 120573
(1)
119895 120573
(119903)
119895)1119902
(1 minus 119889(1)
119895 120575
(1)
119895)11199021
(1 minus 119889(119903)
119895 120575
(119903)
119895)1119902119903
]
]
]
(24)
(v) I001119901111199011199030011990111199021+1119901
119903119902119903+1
[
[
[
minus1199111
minus119911
119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
mdash (1 minus 119888(1)
119895 120574
(1)
119895 1)
11199011
(1 minus 119888(119903)
119895 120574
(119903)
119895 1)
1119901119903
mdash (0 1 1) (1 minus 119889(1)
119895 120575
(1)
119895 1)
11199021
(0 1 1) (1 minus 119889(119903)
119895 120575
(119903)
119895 1)
1119902119903
]
]
]
=1199011
Ψ1199021
[
[
(119888(1)
119895 120574
(1)
119895)11199011
(119889(1)
119895 120575
(1)
119895)11199021
1199111]
]
times sdot sdot sdot times119901119903
Ψ119902119903
[
[
(119888(119903)
119895 120574
(119903)
119895)1119901119903
(119889(119903)
119895 120575
(119903)
119895)1119902119903
119911119903]
]
(25)
6 International Journal of Engineering Mathematics
where the functions119901119894
Ψ119902119894
119894 = 1 119903 are Wrightrsquos general-ized hypergeometric functions [11]
(vi) I001010000202
[
[
[
1199111
119911119903
10038161003816100381610038161003816100381610038161003816
mdash mdash mdashmdash (0 1 1) (minus120583
1 120572
1 1) (0 1 1) (minus120583
119903 120572
119903 1)
]
]
]
=
119903
prod
119894=1
119869120572119894
120583119894
(119911119894) (26)
where the functions 119869120572119894
120583119894
(119911119894) are Wrightrsquos generalized Bessel
functions [12]
(vii) I001212002222
[
[
[
minus1199111
minus119911
119903
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
mdash (1 1 1) (1-1205721 1 120583
1) (1 1 1) (1-120572
119903 1 120583
119903)
mdash (0 1 1) (-1205721 1 120583
1) (0 1 1) (-120572
119903 1 120583
119903)
]
]
]
=
119903
prod
119894=1
Φ(119911119894 120583
119894 120572
119894) (27)
where Φ(119911119894 120583
119894 120572
119894) 119894 = 1 119903 are the generalized Riemann
zeta functions [13 page 27 111 (1)] which are the generaliza-tions of Hurwitz zeta functions and Riemann zeta functions[13 page 24 110 (1) and 112 (1)]
(viii) I001212002222
[
[
[
minus1199111
minus119911
119903
10038161003816100381610038161003816100381610038161003816100381610038161003816
mdash (1 1 1) (1 1 1205831) (1 1 1) (1 1 120583
119903)
mdash (0 1 1) (0 1 1205831) (0 1 1) (0 1 120583
119903)
]
]
]
=
119903
prod
119894=1
119865 (119911119894 120583
119894) (28)
where 119865(119911119894 120583
119894) are the polylogarithms of order 120583
119894 For 120583
119894= 2
119894 = 1 119903 theRHS of (28) reduces to the product of Eulerrsquosdilogarithm [13 page 31 1111 equation (2)]
6 Elementary Properties andTransformation Formulas
Theproperties given below are immediate consequence of thedefinition (1) and hence they are given here without proof
(i) I001198981 119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895119861119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I001198991 1198981119899119903 11989811990311990211990111990211199011119902119903119901119903
[
[
[
119911minus1
1
119911minus1
119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(1 minus 119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(1 minus 119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(1 minus 119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
(1 minus 119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(1 minus 119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(1 minus 119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
]
]
]
(29)
(ii) 1199111198961
1sdot sdot sdot 119911
119896119903
119903I [119911
1sdot sdot sdot 119911
119903]
= I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
[
[
1199111
119911119903
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895+
119903
sum
119894=1
119896119894120572(119894)
119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)
1119901
(119888(1)
119895+119896
1120574(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895+119896
119903120574(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895+
119903
sum
119894=1
119896119894120573(119894)
119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)
1119902
(119889(1)
119895+ 119896
1120575(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895+ 119896
119903120575(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
]
]
(30)
for 119896119894gt 0 119894 = 1 119903
International Journal of Engineering Mathematics 7
(iii) 1
1198961
sdot sdot sdot
1
119896119903
I [1199111 119911
119903]
= I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
[
1199111198961
1
119911119896119903
119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 119896
1120572(1)
119895 119896
119903120572(119903)
119895 119860
119895)1119901
(119888(1)
119895 119896
1120574(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 119896
119903120574(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 119896
1120573(1)
119895 119896
119903120573(119903)
119895 119861
119895)1119901
(119889(1)
119895 119896
1120575(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 119896
119903120575(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
]
(31)
where 119896119894gt 0 119894 = 1 119903
(iv) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886 120572 0 0 119860) (119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I0119899minus111989811198991+1119898119903 1198991199031199011199021199011+11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 )
2119901 (119886 120572 119860) (119888
(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(32)
where 119901 ge 119899 ge 1
(v) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119886 120572 0 0 119860) (119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I011989911989811198991 119898119903 119899119903119901minus1119902119901
1+11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119886 120572 119860) (119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(33)
where 119901 minus 1 ge 119899 ge 0
(vi) I011989911989811198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119887 120573 0 0 119861) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I011989911989811198991 119898119903 119899119903119901119902minus1119901
11199021+1119901
119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119887 120573 119861) (119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(34)
where 119902 minus 1 ge 0
8 International Journal of Engineering Mathematics
(vii) I01198991198981 1198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886 0 0 119860) (119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= Γ119860(1 minus 119886) times I0119899minus11198981 1198991119898119903119899119903
119901minus111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(35)
where 119901 ge 119899 ge 1R(1 minus 119886) gt 0
(viii) I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119886 0 0 119860) (119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119860(119886)
times I011989911989811198991119898119903119899119903119901minus1119902119901
11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(36)
where 119901 minus 1 ge 119899 ge 0R(119886) gt 0
(ix) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119887 0 0 119861) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119861(1 minus 119887)
times I011989911989811198991 119898119903 119899119903119901119902minus1119901
11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(37)
where 119902 minus 1 ge 0R(1 minus 119887) gt 0
(x) I011989911989811198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888 0 119862) (119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= Γ119862(1 minus 119888) times I011989911989811198991minus1119898119903 119899119903
1199011199021199011minus11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(38)
where 1199011ge 119899
1ge 1R(1 minus 119888) gt 0
International Journal of Engineering Mathematics 9
(xi) I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888 0 119862) (119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119862(119888)
times I011989911989811198991 119898119903 1198991199031199011199021199011minus11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(39)
where 1199011minus 1 ge 119899
1ge 0R(119888) gt 0
(xii) I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889 0 119863) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= Γ119863(119889) times I01198991198981minus11198991 119898119903119899119903
11990111990211990111199021minus1119901
119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(40)
where 1199021ge 119898
1ge 1R(119889) gt 0
(xiii) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889 0 119863) (119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119863(1 minus 119889)
times I01198991198981119899111989811990311989911990311990111990211990111199021minus1119901
119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(41)
provided that 1199021minus 1 ge 119898
1ge 0R(1 minus 119889) ge 0
(xiv) I01198991198981 1198991 119898119903 1198991199031199011199021199011 1199021 119901119903 119902119903
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902minus1
(1198861 120572(1)
1 120572
(119903)
1 1198601) (119889
1
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119888(1)
1 120574
(1)
1 119862
(1)
1) (119889
(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
(119888(119903)
1 120574
(119903)
1 119862
(119903)
1)
]]]
]
= I0119899minus11198981 1198991minus1119898119903 119899119903minus1119901minus1119902minus11199011minus11199021minus1119901119903minus1119902119903minus1
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902minus1
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
]]]
]
(42)
provided that 119901 ge 119899 ge 1 119901119894ge 119899
119894ge 1 119894 = 1 119903 and 119902 ge 1
119902119894ge 119898
119894+ 1 119894 = 1 119903
10 International Journal of Engineering Mathematics
(xv) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119889(1)
1 120575
(1)
1 119863
(1)
1) (119888
(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119889(119903)
1 120575
(119903)
1 119863
(119903)
1)
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I01198991198981minus11198991 119898119903minus11198991199031199011199021199011minus11199021minus1119901
119903minus1119902119903minus1
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
(43)
provided that 119901 ge 119899 + 1 119902 ge 1 119901119894ge 119899
119894+ 1 and 119902
119894ge
119898119894ge 1 119894 = 1 119903
(xvi) I011989911989811198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119888(1)
1199011
120574(1)
1199011
119862(1)
1199011
) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119888(119903)
119901119903
120574(119903)
119901119903
119862(119903)
119901119903
) (119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
= I0119899 1198981minus11198991 119898
119903minus1119899119903
119901119902 1199011minus11199021minus1 119901
119903minus1119902119903minus1
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
(44)
provided that 119901119894ge 119899
119894+ 1 119902
119894ge 119898
119894 119894 = 1 119903
(xvii) I01198991198981 1198991 119898119903 1198991199031199011199021199011 1199021 119901119903 119902119903
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119889(1)
1199021 120575
(1)
1199021 119863
(1)
1199021) (119888
(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119889(119903)
119902119903 120575
(119903)
119902119903 119863
(119903)
119902119903) (119888
(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]]]
]
= I01198991198981 1198991minus1119898119903 119899119903minus11199011199021199011minus11199021minus1119901119903minus1119902119903minus1
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
]]]
]
(45)
where 119901 ge 119899 119902 ge 1 119901119894ge 119899
119894ge 1 119902
119894minus 1 ge 119898
119894 119894 = 1 119903
7 Special Cases
When 119903 = 2 and all the exponents 119860119895(119895 = 1 119901)
119861119895(119895 = 1 119902) 119862
(119894)
119895(119895 = 1 119901
119894 119894 = 1 119903) and
119863(119894)
119895(1 119902
119894 119894 = 1 119903) the I-function of ldquo119903rdquo variables
reduces to H-function of two variables and therefore weobtain the corresponding results in H-function of two vari-ables [14]
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgment
The authors are immensely grateful to the worthy referee forsome useful and valuable suggestions for the improvement ofthis paper which led to a better presentation
International Journal of Engineering Mathematics 11
References
[1] A K Rathie ldquoA new generalization of generalized hypergeo-metric functionsrdquo Le Matematiche vol 52 no 2 pp 297ndash3101997
[2] I S Ansari F Yilmaz M S Alouni and O Kucur ldquoNewresults on the sum of Gamma random variates with applicationto the performance of wireless communication systems overnakagami-m fading channelsrdquo httparxiv-web3librarycornelleduabs12022576v4
[3] I S Ansari andYilmaz ldquoOn the sumof squared n-Randomvari-ates with application to the performance of wireless communi-cation systemsrdquo httparxiv-web3librarycornelleduabs12100100v1
[4] X Minghua W Yik-Chung and A Sonia ldquoExact outage prob-ability of dual-hop CSI-assisted AF relaying over nakagami-mfading channelsrdquo IEEE Transactions on Signal Processing vol60 no 10 pp 5578ndash5583 2012
[5] K ShanthaKumari T M Vasudevan Nambisan and AK Rathie ldquoA study of the I-function of two variablesrdquohttparxivorgabs12126717
[6] P K Mittal and K C Gupta ldquoAn integral involving generalizedfunction of two variablesrdquo Proceedings of the Indian Academy ofSciences A vol 75 no 3 pp 117ndash123 1972
[7] HM Srivastava and R Panda ldquoSome bilateral generating func-tions for a class of generalkized hypergeometric polynomialsrdquoJournal Fur die Reine und Angewandte Mathematik vol 17 no288 pp 265ndash274 1976
[8] B L J Braaksma ldquoAsymptotic expansions and analytic continu-ations for a class of Barnesintegralsrdquo Compositio Mathematicahvol 15 pp 239ndash341 1964
[9] Y L LukeThe Special Functions andTheir Approximations vol1 Academic Press New York NY USA 1969
[10] H M Srivastava and M C Daoust ldquoOn Eulerian integralsassociated with Kampe de Ferietrsquos functionrdquo Publications DeLrsquoInstitut Mathematique vol 9 no 23 pp 199ndash202 1969
[11] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Halsted Press Chichester UK
[12] E M Wright ldquoThe asymptotic expansion of the generalizedBessel functionrdquo Proceedings of London Mathematical Societyvol 38 pp 257ndash270 1935
[13] A Erdelyi Higher Transcendental Functions vol 1 McGraw-Hill New York NY USA 1953
[14] H M Srivastava K C Gupta and S P Goyal The HmdashFunctions of One and Two Variables With Applications SouthAsian Publishers New Delhi India 1982
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Engineering Mathematics 3
asymptotic behaviour is based on the following relation forthe gamma function Γ(119911) 119911 = 119909 + 119894119910 119909 119910 isin R [9]
1003816100381610038161003816Γ (119909 + 119894119910)
1003816100381610038161003816sim radic2120587
10038161003816100381610038161199101003816100381610038161003816
119909minus12 exp(minus
1
2
12058710038161003816100381610038161199101003816100381610038161003816)
10038161003816100381610038161199101003816100381610038161003816rarr infin
(8)
Along the contour L119896 if we put 119904
119896= 120590
119896+ 119894119905
119896and take the
limit as |119905119896| rarr infin for 119896 = 1 119903 we obtain by virtue of (8)
that1003816100381610038161003816100381610038161003816100381610038161003816
Γ119860119895(1 minus 119886
119895+
119903
sum
119896=1
120572(119896)
119895119904119896)
1003816100381610038161003816100381610038161003816100381610038161003816
le (2120587)1198601198952(120572
(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816)
[12minusR(119886119895)+120572(119896)
119895120590119896]119860119895
times exp [minus
120587
2
(120572(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816+
10038161003816100381610038161003816I (119886
119895)
10038161003816100381610038161003816) 119860
119895]
(9)
119899
prod
119895=1
1003816100381610038161003816100381610038161003816100381610038161003816
Γ119860119895(1 minus 119886
119895+
119903
sum
119896=1
120572(119896)
119895119904119896)
1003816100381610038161003816100381610038161003816100381610038161003816
le (2120587)sum119899
119895=1(1198601198952)
119899
prod
119895=1
(120572(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816)
[12minusR(119886119895)+120572(119896)
119895120590119896]119860119895
times exp[
[
minus
Π
2
119899
sum
119895=1
(120572(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816+
10038161003816100381610038161003816I (119886
119895)
10038161003816100381610038161003816) 119860
119895]
]
(10)
Similarly we have119901
prod
119895=119899+1
1003816100381610038161003816100381610038161003816100381610038161003816
Γ119860119895(119886
119895minus
119903
sum
119896=1
120572(119896)
119895119904119896)
1003816100381610038161003816100381610038161003816100381610038161003816
ge (2120587)sum119896
119895=119899+1(1198601198952)
119896
prod
119895=119899+1
(120572(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816)
[R(119886119895)minus120572(119896)
119895120590119896minus12]119860
119895
times exp[
[
minus
120587
2
119901
prod
119895=119899+1
(120572(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816+
10038161003816100381610038161003816I (119886
119895)
10038161003816100381610038161003816) 119860
119895]
]
119902
prod
119895=1
1003816100381610038161003816100381610038161003816100381610038161003816
Γ119861119895(1 minus 119887
119895+
119903
sum
119896=1
120573(119896)
119895119904119896)
1003816100381610038161003816100381610038161003816100381610038161003816
ge (2120587)sum119902
119895=1(1198611198952)
119902
prod
119895=1
(120573(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816)
[12minusR(119887119895)+120573(119896)
119895120590119896]119861119895
times exp[
[
minus
Π
2
119902
prod
119895=1
(120573(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816+
10038161003816100381610038161003816I (119886
119895)
10038161003816100381610038161003816) 119861
119895]
]
119899119896
prod
119895=1
100381610038161003816100381610038161003816
Γ119862(119896)
119895(1 minus 119888
(119896)
119895+ 120574
(119896)
119895119904119896)
100381610038161003816100381610038161003816
le (2120587)sum119899119896
119895=1(119862(119896)
1198952)
119899119896
prod
119895=1
(120574(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816)
[12minusR(119888(119896)
119895)+120574(119896)
119895120590119896]119862(119896)
119895
times exp[
[
minus
Π
2
119899119896
prod
119895=1
(120574(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816+
10038161003816100381610038161003816I (119888
(119896)
119895)
10038161003816100381610038161003816) 119862
(119896)
119895]
]
119901119896
prod
119895=119899119896+1
100381610038161003816100381610038161003816
Γ119862(119896)
119895(119888
(119896)
119895minus 120574
(119896)
119895119904119896)
100381610038161003816100381610038161003816
ge (2120587)sum119901119896
119895=119899119896+1(119862(119896)
1198952)
119901119896
prod
119895=119899119896+1
(120574(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816)
[R(119888(119896)
119895)minus120574(119896)
119895120590119896minus12]119862
(119896)
119895
times exp[
[
minus
120587
2
119901119896
prod
119895=119899119896+1
(120574(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816+
10038161003816100381610038161003816I (119888
(119896)
119895)
10038161003816100381610038161003816) 119862
(119896)
119895]
]
119898119896
prod
119895=1
100381610038161003816100381610038161003816
Γ119863(119896)
119895(119889
(119896)
119895minus 120575
(119896)
119895119904119896)
100381610038161003816100381610038161003816
le (2120587)sum119898119896
119895=1(119863(119896)
1198952)
119898119896
prod
119895=1
(120575(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816)
[R(119889(119896)
119895)+120575(119896)
119895120590119896minus12]119863
(119896)
119895
times exp[
[
minus
120587
2
119898119896
prod
119895=1
(120575(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816+
10038161003816100381610038161003816I (119889
(119896)
119895)
10038161003816100381610038161003816)119863
(119896)
119895]
]
119902119896
prod
119895=119898119896+1
100381610038161003816100381610038161003816
Γ119863(119896)
119895(1 minus 119889
(119896)
119895+ 120575
(119896)
119895119904119896)
100381610038161003816100381610038161003816
le (2120587)sum119902119896
119895=119898119896+1(119863(119896)
1198952)
119902119896
prod
119895=119898119896+1
(120575(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816)
[12minusR(119889(119896)
119895)+120575(119896)
119895120590119896]119863(119896)
119895
times exp[
[
minus
120587
2
119902119896
prod
119895=119898119896+1
(120575(119896)
119895
1003816100381610038161003816119905119896
1003816100381610038161003816+
10038161003816100381610038161003816I (119889
(119896)
119895)
10038161003816100381610038161003816)119863
(119896)
119895]
]
(11)
Also
119911119904119896
119896= exp [(120590
119896+ 119894119905
119896) (log 1003816
100381610038161003816119911119896
1003816100381610038161003816+ 119894 arg (119911
119896))]
= exp [120590119896log 1003816
100381610038161003816119911119896
1003816100381610038161003816minus 119905
119896arg (119911
119896)]
=1003816100381610038161003816119911119896
1003816100381610038161003816
120590119896 exp [minus119905
119896arg (119911
119896)]
(12)
Hence substituting (10)-(11) in (1) and using (12) we haveafter much simplification
1003816100381610038161003816120601 (119904
1 119904
119903) 120579
119896(119904119896) 119911
119904119896
119896
1003816100381610038161003816
sim 119862119896
1003816100381610038161003816119905119896
1003816100381610038161003816
Ω119896+120583119896120590119896 exp [minus119905
119896arg (119911
119896) minus
120587
2
1003816100381610038161003816119905119896
1003816100381610038161003816Δ119896]
(13)
where 119862119896is independent of 119905
119896and Δ
119896 120583
119896 and Ω
119896are given
by (6) (7) and (8) respectively for each 119896 = 1 2 119903Hence the result follows
4 International Journal of Engineering Mathematics
Remark 1 If 119863(119894)
119895= 1 (119895 = 1 119898
119894 119894 = 1 119903) in (1) then
the function will be denoted by
I[[
[
1199111
119911119903
]
]
]
= I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
times[
[
[
1199111
119911119903
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 1)
11198981
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)1198981+11199021
(119889(119903)
119895 120575
(119903)
119895 1)
1119898119903
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)119898119903+1119902119903
]
]
]
=
1
(2120587119894)119903int
L1
sdot sdot sdot int
L119903
120601 (1199041 119904
119903) 120579
1(1199041) sdot sdot sdot 120579
119903(119904119903) 119911
1199041
1sdot sdot sdot 119911
119904119903
119903119889119904
1sdot sdot sdot 119889119904
119903
(14)
where
120579119894(119904119894) =
119899119894
prod
119895=1
Γ119862(119894)
119895(1 minus 119888
(119894)
119895+ 120574
(119894)
119895119904119894)
119898119894
prod
119895=1
Γ (119889(119894)
119895minus 120575
(119894)
119895119904119894)
times(
119901119894
prod
119895=119899119894+1
Γ119862(119894)
119895(119888(119894)
119895minus 120574
(119894)
119895119904119894)
119902119894
prod
119895=119898119894+1
Γ119863(119894)
119895(1minus 119889
(119894)
119895+120575
(119894)
119895119904119894))
minus1
(15)where 119894 = 1 119903
Remark 2 If 119862(119894)
119895= 1 (119895 = 1 119899
119894) 119863(119894)
119895= 1 (119895 = 1 119898
119894)
where 119894 = 1 119903 and if 119899 = 0 in (1) then the correspondingfunction will be denoted by
I1
[
[
[
1199111
119911119903
]
]
]
= I001198981 119899111989811990311989911990311990111990211990111199021119901119903119902119903
times[
[
[
1199111
119911119903
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 1)
11198991
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)1198991+11199011
(119888(119903)
119895 120574
(119903)
119895 1)
1119899119903
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)119899119903+1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 1)
11198981
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)1198981+11199021
(119889(119903)
119895 120575
(119903)
119895 1)
1119898119903
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)119898119903+1119902119903
]
]
]
=
1
(2120587119894)119903int
L1
sdot sdot sdot int
L119903
1206011(1199041 119904
119903) 120579
1(1199041) sdot sdot sdot 120579
119903(119904119903) 119911
1199041
1sdot sdot sdot 119911
119904119903
119903119889119904
1sdot sdot sdot 119889119904
119903
(16)
where1206011(1199041 119904
119903)
=
1
prod119901
119895=1Γ119860119895(119886
119895minussum
119903
119894=1120572(119894)
119895119904119894) prod
119902
119895=1Γ119861119895(1minus119887
119895+sum
119903
119894=1120573(119894)
119895119904119894)
120579119894(119904119894) =
prod119899119894
119895=1Γ (1minus119888
(119894)
119895+120574
(119894)
119895119904119894)prod
119898119894
119895=1Γ (119889
(119894)
119895minus 120575
(119894)
119895119904119894)
prod119901119894
119895=119899119894+1
Γ119862(119894)
119895(119888
(119894)
119895minus120574
(119894)
119895119904119894)prod
119902119894
119895=119898119894+1
Γ119863(119894)
119895(1minus119889
(119894)
119895+120575
(119894)
119895119904119894)
forall119894 = 1 119903
(17)
4 Series Representationif
(i) 119911119894
= 0 (119894 = 1 119903) and 120583119894lt 0 where 120583
119894is given by
(4)
(ii) 120575(119894)
ℎ119894
(119889(119894)
119895+119896
119894) = 120575
(119894)
119895(119889
(119894)
ℎ119894
+120578119894) for 119895 = ℎ
119894 119895 ℎ
119894= 1 119898
119894
(119894 = 1 119903) 119896119894 120578
119894= 0 1 2 (119894 = 1 119903)
then
I[[
[
1199111
119911119903
]
]
]
=
1198981
sum
ℎ1=1
sdot sdot sdot
119898119903
sum
ℎ119903=1
infin
sum
1198961=1
sdot sdot sdot
infin
sum
119896119903=1
times [1206011(
119889ℎ(1)
1+ 119896
1
120575ℎ(1)
1
119889ℎ(119903)
119903+ 119896
119903
120575ℎ(119903)
119903
)
times
119903
prod
119894=1
(minus1)119896119894
120575ℎ(119894)
119894119896119894
119911(119889ℎ119894+119896119894)120575ℎ119894
119894]
119895 = ℎ119894
(18)
This result can be proved on computing the residues at the
International Journal of Engineering Mathematics 5
poles as follows
119904119903=
119889ℎ(119894)
119894+ 119896
119894
120575ℎ(119894)
119894
(ℎ119894= 1 119898
119894 119896
119894= 0 1 2 ) for 119894 = 1 119903
(19)
The behaviour of the function I[1199111119911119903
] is given by
I[[
[
1199111
119911119903
]
]
]
= 119874(
119903
prod
119895=1
1003816100381610038161003816119911119894
1003816100381610038161003816
120601119895
) max 10038161003816100381610038161199111
1003816100381610038161003816
1003816100381610038161003816119911119903
1003816100381610038161003816 997888rarr 0
(20)
where
120601119895= min
1le119895le119898119894
[
[
Re(119889(119894)
119895
120575(119894)
119895
)]
]
(119894 = 1 119903) (21)
On the other hand when |119911119894| rarr infin (119894 = 1 119903) the
associated function I1[
1199111119911119903
] given by (16) has the behaviour
I1
[
[
[
1199111
119911119903
]
]
]
= 119874(
119903
prod
119895=1
10038161003816100381610038161003816119911119895
10038161003816100381610038161003816
120601119895
) min 10038161003816100381610038161199111
1003816100381610038161003816
1003816100381610038161003816119911119903
1003816100381610038161003816 997888rarr 0
(22)
where
120601119895= max
1le119895le119899119894
[
[
Re(1 minus 119888
(119894)
119895
120574(119894)
119895
)]
]
(119894 = 1 119903) (23)
5 Elementary Special Cases
In this section we mention some interesting and usefulspecial cases of the I-function of ldquo119903rdquo variables
(i) If all the exponents 119860119895(119895 = 1 119901) 119861
119895(119895 =
1 119902) 119862(119894)
119895(119895 = 1 119901
119894 119894 = 1 119903) and
119863(119894)
119895(119895 = 1 119902
119894 119894 = 1 119903) in (1) are equal to
unity we obtain H-function of ldquo119903rdquo variables definedby Srivastava and Panda [7]
(ii) When 119901 = 119902 = 119899 = 0 (1) degenerates into the productof 119903mutually independent I- functions of one variableintroduced by Rathie [1]
(iii) When 119901 = 119902 = 119899 = 0 and 119903 = 1 (1) reduces to theI-function defined by Rathie [1]
(iv) When 119899 = 119901 119898119894= 1 119899
119894= 119901
119894 119894 = 1 119903 and 119860
119895=
119861119895= 119862
119895= 119863
119895= 1 and (119889
(119894)
119895 120575
(119894)
119895 119863
(119894)
119895) is replaced by
(0 1 1) (119889(119894)119895 120575
(119894)
119895 119863
(119894)
119895) (1) reduces to the generalized
Lauricella function [10]
I011990111199011111990111990311990111990211990111199021+1119901
119903119902119903+1
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 1)
1119901 (119888
(1)
119895 120574
(1)
119895 1)
11199011
(119888(119903)
119895 120574
(119903)
119895 1)
1119901119903
(119887119895 120573
(1)
119895 120573
(1)
119895 120573
(119903)
119895 1)
1119902 (0 1 1) (119889
(1)
119895 120575
(1)
119895 1)
11199021
(0 1 1) (119889(119903)
119895 120575
(119903)
119895 1)
1119902119903
]
]
]
=
prod119901
119895=1Γ (1 minus 119886
119895)prod
1199011
119895=1Γ (1 minus 119888
(1)
119895) sdot sdot sdotprod
119901119903
119895=1Γ (1 minus 119888
(119903)
119895)
prod119902
119895=1Γ (1 minus 119887
119895)prod
1199021
119895=1Γ (1 minus 119889
(1)
119895) sdot sdot sdotprod
119902119903
119895=1Γ (1 minus 119889
(119903)
119895)
times 1198651199011199011119901119903
1199021199021119902119903
[
[
[
minus1199111
minus119911
119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(1 minus 119886119895 120572
(1)
119895 120572
(119903)
119895)1119901
(1 minus 119888(1)
119895 120574
(1)
119895)11199011
(1 minus 119888((119903))
119895 120574
(119903)
119895)1119901119903
(1 minus 119887119895 120573
(1)
119895 120573
(119903)
119895)1119902
(1 minus 119889(1)
119895 120575
(1)
119895)11199021
(1 minus 119889(119903)
119895 120575
(119903)
119895)1119902119903
]
]
]
(24)
(v) I001119901111199011199030011990111199021+1119901
119903119902119903+1
[
[
[
minus1199111
minus119911
119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
mdash (1 minus 119888(1)
119895 120574
(1)
119895 1)
11199011
(1 minus 119888(119903)
119895 120574
(119903)
119895 1)
1119901119903
mdash (0 1 1) (1 minus 119889(1)
119895 120575
(1)
119895 1)
11199021
(0 1 1) (1 minus 119889(119903)
119895 120575
(119903)
119895 1)
1119902119903
]
]
]
=1199011
Ψ1199021
[
[
(119888(1)
119895 120574
(1)
119895)11199011
(119889(1)
119895 120575
(1)
119895)11199021
1199111]
]
times sdot sdot sdot times119901119903
Ψ119902119903
[
[
(119888(119903)
119895 120574
(119903)
119895)1119901119903
(119889(119903)
119895 120575
(119903)
119895)1119902119903
119911119903]
]
(25)
6 International Journal of Engineering Mathematics
where the functions119901119894
Ψ119902119894
119894 = 1 119903 are Wrightrsquos general-ized hypergeometric functions [11]
(vi) I001010000202
[
[
[
1199111
119911119903
10038161003816100381610038161003816100381610038161003816
mdash mdash mdashmdash (0 1 1) (minus120583
1 120572
1 1) (0 1 1) (minus120583
119903 120572
119903 1)
]
]
]
=
119903
prod
119894=1
119869120572119894
120583119894
(119911119894) (26)
where the functions 119869120572119894
120583119894
(119911119894) are Wrightrsquos generalized Bessel
functions [12]
(vii) I001212002222
[
[
[
minus1199111
minus119911
119903
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
mdash (1 1 1) (1-1205721 1 120583
1) (1 1 1) (1-120572
119903 1 120583
119903)
mdash (0 1 1) (-1205721 1 120583
1) (0 1 1) (-120572
119903 1 120583
119903)
]
]
]
=
119903
prod
119894=1
Φ(119911119894 120583
119894 120572
119894) (27)
where Φ(119911119894 120583
119894 120572
119894) 119894 = 1 119903 are the generalized Riemann
zeta functions [13 page 27 111 (1)] which are the generaliza-tions of Hurwitz zeta functions and Riemann zeta functions[13 page 24 110 (1) and 112 (1)]
(viii) I001212002222
[
[
[
minus1199111
minus119911
119903
10038161003816100381610038161003816100381610038161003816100381610038161003816
mdash (1 1 1) (1 1 1205831) (1 1 1) (1 1 120583
119903)
mdash (0 1 1) (0 1 1205831) (0 1 1) (0 1 120583
119903)
]
]
]
=
119903
prod
119894=1
119865 (119911119894 120583
119894) (28)
where 119865(119911119894 120583
119894) are the polylogarithms of order 120583
119894 For 120583
119894= 2
119894 = 1 119903 theRHS of (28) reduces to the product of Eulerrsquosdilogarithm [13 page 31 1111 equation (2)]
6 Elementary Properties andTransformation Formulas
Theproperties given below are immediate consequence of thedefinition (1) and hence they are given here without proof
(i) I001198981 119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895119861119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I001198991 1198981119899119903 11989811990311990211990111990211199011119902119903119901119903
[
[
[
119911minus1
1
119911minus1
119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(1 minus 119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(1 minus 119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(1 minus 119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
(1 minus 119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(1 minus 119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(1 minus 119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
]
]
]
(29)
(ii) 1199111198961
1sdot sdot sdot 119911
119896119903
119903I [119911
1sdot sdot sdot 119911
119903]
= I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
[
[
1199111
119911119903
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895+
119903
sum
119894=1
119896119894120572(119894)
119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)
1119901
(119888(1)
119895+119896
1120574(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895+119896
119903120574(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895+
119903
sum
119894=1
119896119894120573(119894)
119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)
1119902
(119889(1)
119895+ 119896
1120575(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895+ 119896
119903120575(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
]
]
(30)
for 119896119894gt 0 119894 = 1 119903
International Journal of Engineering Mathematics 7
(iii) 1
1198961
sdot sdot sdot
1
119896119903
I [1199111 119911
119903]
= I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
[
1199111198961
1
119911119896119903
119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 119896
1120572(1)
119895 119896
119903120572(119903)
119895 119860
119895)1119901
(119888(1)
119895 119896
1120574(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 119896
119903120574(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 119896
1120573(1)
119895 119896
119903120573(119903)
119895 119861
119895)1119901
(119889(1)
119895 119896
1120575(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 119896
119903120575(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
]
(31)
where 119896119894gt 0 119894 = 1 119903
(iv) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886 120572 0 0 119860) (119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I0119899minus111989811198991+1119898119903 1198991199031199011199021199011+11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 )
2119901 (119886 120572 119860) (119888
(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(32)
where 119901 ge 119899 ge 1
(v) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119886 120572 0 0 119860) (119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I011989911989811198991 119898119903 119899119903119901minus1119902119901
1+11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119886 120572 119860) (119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(33)
where 119901 minus 1 ge 119899 ge 0
(vi) I011989911989811198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119887 120573 0 0 119861) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I011989911989811198991 119898119903 119899119903119901119902minus1119901
11199021+1119901
119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119887 120573 119861) (119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(34)
where 119902 minus 1 ge 0
8 International Journal of Engineering Mathematics
(vii) I01198991198981 1198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886 0 0 119860) (119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= Γ119860(1 minus 119886) times I0119899minus11198981 1198991119898119903119899119903
119901minus111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(35)
where 119901 ge 119899 ge 1R(1 minus 119886) gt 0
(viii) I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119886 0 0 119860) (119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119860(119886)
times I011989911989811198991119898119903119899119903119901minus1119902119901
11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(36)
where 119901 minus 1 ge 119899 ge 0R(119886) gt 0
(ix) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119887 0 0 119861) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119861(1 minus 119887)
times I011989911989811198991 119898119903 119899119903119901119902minus1119901
11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(37)
where 119902 minus 1 ge 0R(1 minus 119887) gt 0
(x) I011989911989811198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888 0 119862) (119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= Γ119862(1 minus 119888) times I011989911989811198991minus1119898119903 119899119903
1199011199021199011minus11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(38)
where 1199011ge 119899
1ge 1R(1 minus 119888) gt 0
International Journal of Engineering Mathematics 9
(xi) I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888 0 119862) (119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119862(119888)
times I011989911989811198991 119898119903 1198991199031199011199021199011minus11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(39)
where 1199011minus 1 ge 119899
1ge 0R(119888) gt 0
(xii) I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889 0 119863) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= Γ119863(119889) times I01198991198981minus11198991 119898119903119899119903
11990111990211990111199021minus1119901
119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(40)
where 1199021ge 119898
1ge 1R(119889) gt 0
(xiii) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889 0 119863) (119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119863(1 minus 119889)
times I01198991198981119899111989811990311989911990311990111990211990111199021minus1119901
119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(41)
provided that 1199021minus 1 ge 119898
1ge 0R(1 minus 119889) ge 0
(xiv) I01198991198981 1198991 119898119903 1198991199031199011199021199011 1199021 119901119903 119902119903
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902minus1
(1198861 120572(1)
1 120572
(119903)
1 1198601) (119889
1
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119888(1)
1 120574
(1)
1 119862
(1)
1) (119889
(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
(119888(119903)
1 120574
(119903)
1 119862
(119903)
1)
]]]
]
= I0119899minus11198981 1198991minus1119898119903 119899119903minus1119901minus1119902minus11199011minus11199021minus1119901119903minus1119902119903minus1
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902minus1
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
]]]
]
(42)
provided that 119901 ge 119899 ge 1 119901119894ge 119899
119894ge 1 119894 = 1 119903 and 119902 ge 1
119902119894ge 119898
119894+ 1 119894 = 1 119903
10 International Journal of Engineering Mathematics
(xv) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119889(1)
1 120575
(1)
1 119863
(1)
1) (119888
(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119889(119903)
1 120575
(119903)
1 119863
(119903)
1)
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I01198991198981minus11198991 119898119903minus11198991199031199011199021199011minus11199021minus1119901
119903minus1119902119903minus1
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
(43)
provided that 119901 ge 119899 + 1 119902 ge 1 119901119894ge 119899
119894+ 1 and 119902
119894ge
119898119894ge 1 119894 = 1 119903
(xvi) I011989911989811198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119888(1)
1199011
120574(1)
1199011
119862(1)
1199011
) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119888(119903)
119901119903
120574(119903)
119901119903
119862(119903)
119901119903
) (119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
= I0119899 1198981minus11198991 119898
119903minus1119899119903
119901119902 1199011minus11199021minus1 119901
119903minus1119902119903minus1
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
(44)
provided that 119901119894ge 119899
119894+ 1 119902
119894ge 119898
119894 119894 = 1 119903
(xvii) I01198991198981 1198991 119898119903 1198991199031199011199021199011 1199021 119901119903 119902119903
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119889(1)
1199021 120575
(1)
1199021 119863
(1)
1199021) (119888
(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119889(119903)
119902119903 120575
(119903)
119902119903 119863
(119903)
119902119903) (119888
(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]]]
]
= I01198991198981 1198991minus1119898119903 119899119903minus11199011199021199011minus11199021minus1119901119903minus1119902119903minus1
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
]]]
]
(45)
where 119901 ge 119899 119902 ge 1 119901119894ge 119899
119894ge 1 119902
119894minus 1 ge 119898
119894 119894 = 1 119903
7 Special Cases
When 119903 = 2 and all the exponents 119860119895(119895 = 1 119901)
119861119895(119895 = 1 119902) 119862
(119894)
119895(119895 = 1 119901
119894 119894 = 1 119903) and
119863(119894)
119895(1 119902
119894 119894 = 1 119903) the I-function of ldquo119903rdquo variables
reduces to H-function of two variables and therefore weobtain the corresponding results in H-function of two vari-ables [14]
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgment
The authors are immensely grateful to the worthy referee forsome useful and valuable suggestions for the improvement ofthis paper which led to a better presentation
International Journal of Engineering Mathematics 11
References
[1] A K Rathie ldquoA new generalization of generalized hypergeo-metric functionsrdquo Le Matematiche vol 52 no 2 pp 297ndash3101997
[2] I S Ansari F Yilmaz M S Alouni and O Kucur ldquoNewresults on the sum of Gamma random variates with applicationto the performance of wireless communication systems overnakagami-m fading channelsrdquo httparxiv-web3librarycornelleduabs12022576v4
[3] I S Ansari andYilmaz ldquoOn the sumof squared n-Randomvari-ates with application to the performance of wireless communi-cation systemsrdquo httparxiv-web3librarycornelleduabs12100100v1
[4] X Minghua W Yik-Chung and A Sonia ldquoExact outage prob-ability of dual-hop CSI-assisted AF relaying over nakagami-mfading channelsrdquo IEEE Transactions on Signal Processing vol60 no 10 pp 5578ndash5583 2012
[5] K ShanthaKumari T M Vasudevan Nambisan and AK Rathie ldquoA study of the I-function of two variablesrdquohttparxivorgabs12126717
[6] P K Mittal and K C Gupta ldquoAn integral involving generalizedfunction of two variablesrdquo Proceedings of the Indian Academy ofSciences A vol 75 no 3 pp 117ndash123 1972
[7] HM Srivastava and R Panda ldquoSome bilateral generating func-tions for a class of generalkized hypergeometric polynomialsrdquoJournal Fur die Reine und Angewandte Mathematik vol 17 no288 pp 265ndash274 1976
[8] B L J Braaksma ldquoAsymptotic expansions and analytic continu-ations for a class of Barnesintegralsrdquo Compositio Mathematicahvol 15 pp 239ndash341 1964
[9] Y L LukeThe Special Functions andTheir Approximations vol1 Academic Press New York NY USA 1969
[10] H M Srivastava and M C Daoust ldquoOn Eulerian integralsassociated with Kampe de Ferietrsquos functionrdquo Publications DeLrsquoInstitut Mathematique vol 9 no 23 pp 199ndash202 1969
[11] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Halsted Press Chichester UK
[12] E M Wright ldquoThe asymptotic expansion of the generalizedBessel functionrdquo Proceedings of London Mathematical Societyvol 38 pp 257ndash270 1935
[13] A Erdelyi Higher Transcendental Functions vol 1 McGraw-Hill New York NY USA 1953
[14] H M Srivastava K C Gupta and S P Goyal The HmdashFunctions of One and Two Variables With Applications SouthAsian Publishers New Delhi India 1982
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 International Journal of Engineering Mathematics
Remark 1 If 119863(119894)
119895= 1 (119895 = 1 119898
119894 119894 = 1 119903) in (1) then
the function will be denoted by
I[[
[
1199111
119911119903
]
]
]
= I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
times[
[
[
1199111
119911119903
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 1)
11198981
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)1198981+11199021
(119889(119903)
119895 120575
(119903)
119895 1)
1119898119903
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)119898119903+1119902119903
]
]
]
=
1
(2120587119894)119903int
L1
sdot sdot sdot int
L119903
120601 (1199041 119904
119903) 120579
1(1199041) sdot sdot sdot 120579
119903(119904119903) 119911
1199041
1sdot sdot sdot 119911
119904119903
119903119889119904
1sdot sdot sdot 119889119904
119903
(14)
where
120579119894(119904119894) =
119899119894
prod
119895=1
Γ119862(119894)
119895(1 minus 119888
(119894)
119895+ 120574
(119894)
119895119904119894)
119898119894
prod
119895=1
Γ (119889(119894)
119895minus 120575
(119894)
119895119904119894)
times(
119901119894
prod
119895=119899119894+1
Γ119862(119894)
119895(119888(119894)
119895minus 120574
(119894)
119895119904119894)
119902119894
prod
119895=119898119894+1
Γ119863(119894)
119895(1minus 119889
(119894)
119895+120575
(119894)
119895119904119894))
minus1
(15)where 119894 = 1 119903
Remark 2 If 119862(119894)
119895= 1 (119895 = 1 119899
119894) 119863(119894)
119895= 1 (119895 = 1 119898
119894)
where 119894 = 1 119903 and if 119899 = 0 in (1) then the correspondingfunction will be denoted by
I1
[
[
[
1199111
119911119903
]
]
]
= I001198981 119899111989811990311989911990311990111990211990111199021119901119903119902119903
times[
[
[
1199111
119911119903
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 1)
11198991
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)1198991+11199011
(119888(119903)
119895 120574
(119903)
119895 1)
1119899119903
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)119899119903+1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 1)
11198981
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)1198981+11199021
(119889(119903)
119895 120575
(119903)
119895 1)
1119898119903
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)119898119903+1119902119903
]
]
]
=
1
(2120587119894)119903int
L1
sdot sdot sdot int
L119903
1206011(1199041 119904
119903) 120579
1(1199041) sdot sdot sdot 120579
119903(119904119903) 119911
1199041
1sdot sdot sdot 119911
119904119903
119903119889119904
1sdot sdot sdot 119889119904
119903
(16)
where1206011(1199041 119904
119903)
=
1
prod119901
119895=1Γ119860119895(119886
119895minussum
119903
119894=1120572(119894)
119895119904119894) prod
119902
119895=1Γ119861119895(1minus119887
119895+sum
119903
119894=1120573(119894)
119895119904119894)
120579119894(119904119894) =
prod119899119894
119895=1Γ (1minus119888
(119894)
119895+120574
(119894)
119895119904119894)prod
119898119894
119895=1Γ (119889
(119894)
119895minus 120575
(119894)
119895119904119894)
prod119901119894
119895=119899119894+1
Γ119862(119894)
119895(119888
(119894)
119895minus120574
(119894)
119895119904119894)prod
119902119894
119895=119898119894+1
Γ119863(119894)
119895(1minus119889
(119894)
119895+120575
(119894)
119895119904119894)
forall119894 = 1 119903
(17)
4 Series Representationif
(i) 119911119894
= 0 (119894 = 1 119903) and 120583119894lt 0 where 120583
119894is given by
(4)
(ii) 120575(119894)
ℎ119894
(119889(119894)
119895+119896
119894) = 120575
(119894)
119895(119889
(119894)
ℎ119894
+120578119894) for 119895 = ℎ
119894 119895 ℎ
119894= 1 119898
119894
(119894 = 1 119903) 119896119894 120578
119894= 0 1 2 (119894 = 1 119903)
then
I[[
[
1199111
119911119903
]
]
]
=
1198981
sum
ℎ1=1
sdot sdot sdot
119898119903
sum
ℎ119903=1
infin
sum
1198961=1
sdot sdot sdot
infin
sum
119896119903=1
times [1206011(
119889ℎ(1)
1+ 119896
1
120575ℎ(1)
1
119889ℎ(119903)
119903+ 119896
119903
120575ℎ(119903)
119903
)
times
119903
prod
119894=1
(minus1)119896119894
120575ℎ(119894)
119894119896119894
119911(119889ℎ119894+119896119894)120575ℎ119894
119894]
119895 = ℎ119894
(18)
This result can be proved on computing the residues at the
International Journal of Engineering Mathematics 5
poles as follows
119904119903=
119889ℎ(119894)
119894+ 119896
119894
120575ℎ(119894)
119894
(ℎ119894= 1 119898
119894 119896
119894= 0 1 2 ) for 119894 = 1 119903
(19)
The behaviour of the function I[1199111119911119903
] is given by
I[[
[
1199111
119911119903
]
]
]
= 119874(
119903
prod
119895=1
1003816100381610038161003816119911119894
1003816100381610038161003816
120601119895
) max 10038161003816100381610038161199111
1003816100381610038161003816
1003816100381610038161003816119911119903
1003816100381610038161003816 997888rarr 0
(20)
where
120601119895= min
1le119895le119898119894
[
[
Re(119889(119894)
119895
120575(119894)
119895
)]
]
(119894 = 1 119903) (21)
On the other hand when |119911119894| rarr infin (119894 = 1 119903) the
associated function I1[
1199111119911119903
] given by (16) has the behaviour
I1
[
[
[
1199111
119911119903
]
]
]
= 119874(
119903
prod
119895=1
10038161003816100381610038161003816119911119895
10038161003816100381610038161003816
120601119895
) min 10038161003816100381610038161199111
1003816100381610038161003816
1003816100381610038161003816119911119903
1003816100381610038161003816 997888rarr 0
(22)
where
120601119895= max
1le119895le119899119894
[
[
Re(1 minus 119888
(119894)
119895
120574(119894)
119895
)]
]
(119894 = 1 119903) (23)
5 Elementary Special Cases
In this section we mention some interesting and usefulspecial cases of the I-function of ldquo119903rdquo variables
(i) If all the exponents 119860119895(119895 = 1 119901) 119861
119895(119895 =
1 119902) 119862(119894)
119895(119895 = 1 119901
119894 119894 = 1 119903) and
119863(119894)
119895(119895 = 1 119902
119894 119894 = 1 119903) in (1) are equal to
unity we obtain H-function of ldquo119903rdquo variables definedby Srivastava and Panda [7]
(ii) When 119901 = 119902 = 119899 = 0 (1) degenerates into the productof 119903mutually independent I- functions of one variableintroduced by Rathie [1]
(iii) When 119901 = 119902 = 119899 = 0 and 119903 = 1 (1) reduces to theI-function defined by Rathie [1]
(iv) When 119899 = 119901 119898119894= 1 119899
119894= 119901
119894 119894 = 1 119903 and 119860
119895=
119861119895= 119862
119895= 119863
119895= 1 and (119889
(119894)
119895 120575
(119894)
119895 119863
(119894)
119895) is replaced by
(0 1 1) (119889(119894)119895 120575
(119894)
119895 119863
(119894)
119895) (1) reduces to the generalized
Lauricella function [10]
I011990111199011111990111990311990111990211990111199021+1119901
119903119902119903+1
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 1)
1119901 (119888
(1)
119895 120574
(1)
119895 1)
11199011
(119888(119903)
119895 120574
(119903)
119895 1)
1119901119903
(119887119895 120573
(1)
119895 120573
(1)
119895 120573
(119903)
119895 1)
1119902 (0 1 1) (119889
(1)
119895 120575
(1)
119895 1)
11199021
(0 1 1) (119889(119903)
119895 120575
(119903)
119895 1)
1119902119903
]
]
]
=
prod119901
119895=1Γ (1 minus 119886
119895)prod
1199011
119895=1Γ (1 minus 119888
(1)
119895) sdot sdot sdotprod
119901119903
119895=1Γ (1 minus 119888
(119903)
119895)
prod119902
119895=1Γ (1 minus 119887
119895)prod
1199021
119895=1Γ (1 minus 119889
(1)
119895) sdot sdot sdotprod
119902119903
119895=1Γ (1 minus 119889
(119903)
119895)
times 1198651199011199011119901119903
1199021199021119902119903
[
[
[
minus1199111
minus119911
119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(1 minus 119886119895 120572
(1)
119895 120572
(119903)
119895)1119901
(1 minus 119888(1)
119895 120574
(1)
119895)11199011
(1 minus 119888((119903))
119895 120574
(119903)
119895)1119901119903
(1 minus 119887119895 120573
(1)
119895 120573
(119903)
119895)1119902
(1 minus 119889(1)
119895 120575
(1)
119895)11199021
(1 minus 119889(119903)
119895 120575
(119903)
119895)1119902119903
]
]
]
(24)
(v) I001119901111199011199030011990111199021+1119901
119903119902119903+1
[
[
[
minus1199111
minus119911
119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
mdash (1 minus 119888(1)
119895 120574
(1)
119895 1)
11199011
(1 minus 119888(119903)
119895 120574
(119903)
119895 1)
1119901119903
mdash (0 1 1) (1 minus 119889(1)
119895 120575
(1)
119895 1)
11199021
(0 1 1) (1 minus 119889(119903)
119895 120575
(119903)
119895 1)
1119902119903
]
]
]
=1199011
Ψ1199021
[
[
(119888(1)
119895 120574
(1)
119895)11199011
(119889(1)
119895 120575
(1)
119895)11199021
1199111]
]
times sdot sdot sdot times119901119903
Ψ119902119903
[
[
(119888(119903)
119895 120574
(119903)
119895)1119901119903
(119889(119903)
119895 120575
(119903)
119895)1119902119903
119911119903]
]
(25)
6 International Journal of Engineering Mathematics
where the functions119901119894
Ψ119902119894
119894 = 1 119903 are Wrightrsquos general-ized hypergeometric functions [11]
(vi) I001010000202
[
[
[
1199111
119911119903
10038161003816100381610038161003816100381610038161003816
mdash mdash mdashmdash (0 1 1) (minus120583
1 120572
1 1) (0 1 1) (minus120583
119903 120572
119903 1)
]
]
]
=
119903
prod
119894=1
119869120572119894
120583119894
(119911119894) (26)
where the functions 119869120572119894
120583119894
(119911119894) are Wrightrsquos generalized Bessel
functions [12]
(vii) I001212002222
[
[
[
minus1199111
minus119911
119903
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
mdash (1 1 1) (1-1205721 1 120583
1) (1 1 1) (1-120572
119903 1 120583
119903)
mdash (0 1 1) (-1205721 1 120583
1) (0 1 1) (-120572
119903 1 120583
119903)
]
]
]
=
119903
prod
119894=1
Φ(119911119894 120583
119894 120572
119894) (27)
where Φ(119911119894 120583
119894 120572
119894) 119894 = 1 119903 are the generalized Riemann
zeta functions [13 page 27 111 (1)] which are the generaliza-tions of Hurwitz zeta functions and Riemann zeta functions[13 page 24 110 (1) and 112 (1)]
(viii) I001212002222
[
[
[
minus1199111
minus119911
119903
10038161003816100381610038161003816100381610038161003816100381610038161003816
mdash (1 1 1) (1 1 1205831) (1 1 1) (1 1 120583
119903)
mdash (0 1 1) (0 1 1205831) (0 1 1) (0 1 120583
119903)
]
]
]
=
119903
prod
119894=1
119865 (119911119894 120583
119894) (28)
where 119865(119911119894 120583
119894) are the polylogarithms of order 120583
119894 For 120583
119894= 2
119894 = 1 119903 theRHS of (28) reduces to the product of Eulerrsquosdilogarithm [13 page 31 1111 equation (2)]
6 Elementary Properties andTransformation Formulas
Theproperties given below are immediate consequence of thedefinition (1) and hence they are given here without proof
(i) I001198981 119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895119861119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I001198991 1198981119899119903 11989811990311990211990111990211199011119902119903119901119903
[
[
[
119911minus1
1
119911minus1
119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(1 minus 119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(1 minus 119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(1 minus 119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
(1 minus 119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(1 minus 119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(1 minus 119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
]
]
]
(29)
(ii) 1199111198961
1sdot sdot sdot 119911
119896119903
119903I [119911
1sdot sdot sdot 119911
119903]
= I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
[
[
1199111
119911119903
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895+
119903
sum
119894=1
119896119894120572(119894)
119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)
1119901
(119888(1)
119895+119896
1120574(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895+119896
119903120574(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895+
119903
sum
119894=1
119896119894120573(119894)
119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)
1119902
(119889(1)
119895+ 119896
1120575(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895+ 119896
119903120575(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
]
]
(30)
for 119896119894gt 0 119894 = 1 119903
International Journal of Engineering Mathematics 7
(iii) 1
1198961
sdot sdot sdot
1
119896119903
I [1199111 119911
119903]
= I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
[
1199111198961
1
119911119896119903
119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 119896
1120572(1)
119895 119896
119903120572(119903)
119895 119860
119895)1119901
(119888(1)
119895 119896
1120574(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 119896
119903120574(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 119896
1120573(1)
119895 119896
119903120573(119903)
119895 119861
119895)1119901
(119889(1)
119895 119896
1120575(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 119896
119903120575(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
]
(31)
where 119896119894gt 0 119894 = 1 119903
(iv) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886 120572 0 0 119860) (119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I0119899minus111989811198991+1119898119903 1198991199031199011199021199011+11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 )
2119901 (119886 120572 119860) (119888
(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(32)
where 119901 ge 119899 ge 1
(v) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119886 120572 0 0 119860) (119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I011989911989811198991 119898119903 119899119903119901minus1119902119901
1+11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119886 120572 119860) (119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(33)
where 119901 minus 1 ge 119899 ge 0
(vi) I011989911989811198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119887 120573 0 0 119861) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I011989911989811198991 119898119903 119899119903119901119902minus1119901
11199021+1119901
119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119887 120573 119861) (119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(34)
where 119902 minus 1 ge 0
8 International Journal of Engineering Mathematics
(vii) I01198991198981 1198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886 0 0 119860) (119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= Γ119860(1 minus 119886) times I0119899minus11198981 1198991119898119903119899119903
119901minus111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(35)
where 119901 ge 119899 ge 1R(1 minus 119886) gt 0
(viii) I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119886 0 0 119860) (119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119860(119886)
times I011989911989811198991119898119903119899119903119901minus1119902119901
11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(36)
where 119901 minus 1 ge 119899 ge 0R(119886) gt 0
(ix) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119887 0 0 119861) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119861(1 minus 119887)
times I011989911989811198991 119898119903 119899119903119901119902minus1119901
11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(37)
where 119902 minus 1 ge 0R(1 minus 119887) gt 0
(x) I011989911989811198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888 0 119862) (119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= Γ119862(1 minus 119888) times I011989911989811198991minus1119898119903 119899119903
1199011199021199011minus11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(38)
where 1199011ge 119899
1ge 1R(1 minus 119888) gt 0
International Journal of Engineering Mathematics 9
(xi) I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888 0 119862) (119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119862(119888)
times I011989911989811198991 119898119903 1198991199031199011199021199011minus11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(39)
where 1199011minus 1 ge 119899
1ge 0R(119888) gt 0
(xii) I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889 0 119863) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= Γ119863(119889) times I01198991198981minus11198991 119898119903119899119903
11990111990211990111199021minus1119901
119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(40)
where 1199021ge 119898
1ge 1R(119889) gt 0
(xiii) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889 0 119863) (119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119863(1 minus 119889)
times I01198991198981119899111989811990311989911990311990111990211990111199021minus1119901
119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(41)
provided that 1199021minus 1 ge 119898
1ge 0R(1 minus 119889) ge 0
(xiv) I01198991198981 1198991 119898119903 1198991199031199011199021199011 1199021 119901119903 119902119903
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902minus1
(1198861 120572(1)
1 120572
(119903)
1 1198601) (119889
1
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119888(1)
1 120574
(1)
1 119862
(1)
1) (119889
(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
(119888(119903)
1 120574
(119903)
1 119862
(119903)
1)
]]]
]
= I0119899minus11198981 1198991minus1119898119903 119899119903minus1119901minus1119902minus11199011minus11199021minus1119901119903minus1119902119903minus1
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902minus1
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
]]]
]
(42)
provided that 119901 ge 119899 ge 1 119901119894ge 119899
119894ge 1 119894 = 1 119903 and 119902 ge 1
119902119894ge 119898
119894+ 1 119894 = 1 119903
10 International Journal of Engineering Mathematics
(xv) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119889(1)
1 120575
(1)
1 119863
(1)
1) (119888
(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119889(119903)
1 120575
(119903)
1 119863
(119903)
1)
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I01198991198981minus11198991 119898119903minus11198991199031199011199021199011minus11199021minus1119901
119903minus1119902119903minus1
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
(43)
provided that 119901 ge 119899 + 1 119902 ge 1 119901119894ge 119899
119894+ 1 and 119902
119894ge
119898119894ge 1 119894 = 1 119903
(xvi) I011989911989811198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119888(1)
1199011
120574(1)
1199011
119862(1)
1199011
) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119888(119903)
119901119903
120574(119903)
119901119903
119862(119903)
119901119903
) (119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
= I0119899 1198981minus11198991 119898
119903minus1119899119903
119901119902 1199011minus11199021minus1 119901
119903minus1119902119903minus1
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
(44)
provided that 119901119894ge 119899
119894+ 1 119902
119894ge 119898
119894 119894 = 1 119903
(xvii) I01198991198981 1198991 119898119903 1198991199031199011199021199011 1199021 119901119903 119902119903
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119889(1)
1199021 120575
(1)
1199021 119863
(1)
1199021) (119888
(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119889(119903)
119902119903 120575
(119903)
119902119903 119863
(119903)
119902119903) (119888
(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]]]
]
= I01198991198981 1198991minus1119898119903 119899119903minus11199011199021199011minus11199021minus1119901119903minus1119902119903minus1
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
]]]
]
(45)
where 119901 ge 119899 119902 ge 1 119901119894ge 119899
119894ge 1 119902
119894minus 1 ge 119898
119894 119894 = 1 119903
7 Special Cases
When 119903 = 2 and all the exponents 119860119895(119895 = 1 119901)
119861119895(119895 = 1 119902) 119862
(119894)
119895(119895 = 1 119901
119894 119894 = 1 119903) and
119863(119894)
119895(1 119902
119894 119894 = 1 119903) the I-function of ldquo119903rdquo variables
reduces to H-function of two variables and therefore weobtain the corresponding results in H-function of two vari-ables [14]
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgment
The authors are immensely grateful to the worthy referee forsome useful and valuable suggestions for the improvement ofthis paper which led to a better presentation
International Journal of Engineering Mathematics 11
References
[1] A K Rathie ldquoA new generalization of generalized hypergeo-metric functionsrdquo Le Matematiche vol 52 no 2 pp 297ndash3101997
[2] I S Ansari F Yilmaz M S Alouni and O Kucur ldquoNewresults on the sum of Gamma random variates with applicationto the performance of wireless communication systems overnakagami-m fading channelsrdquo httparxiv-web3librarycornelleduabs12022576v4
[3] I S Ansari andYilmaz ldquoOn the sumof squared n-Randomvari-ates with application to the performance of wireless communi-cation systemsrdquo httparxiv-web3librarycornelleduabs12100100v1
[4] X Minghua W Yik-Chung and A Sonia ldquoExact outage prob-ability of dual-hop CSI-assisted AF relaying over nakagami-mfading channelsrdquo IEEE Transactions on Signal Processing vol60 no 10 pp 5578ndash5583 2012
[5] K ShanthaKumari T M Vasudevan Nambisan and AK Rathie ldquoA study of the I-function of two variablesrdquohttparxivorgabs12126717
[6] P K Mittal and K C Gupta ldquoAn integral involving generalizedfunction of two variablesrdquo Proceedings of the Indian Academy ofSciences A vol 75 no 3 pp 117ndash123 1972
[7] HM Srivastava and R Panda ldquoSome bilateral generating func-tions for a class of generalkized hypergeometric polynomialsrdquoJournal Fur die Reine und Angewandte Mathematik vol 17 no288 pp 265ndash274 1976
[8] B L J Braaksma ldquoAsymptotic expansions and analytic continu-ations for a class of Barnesintegralsrdquo Compositio Mathematicahvol 15 pp 239ndash341 1964
[9] Y L LukeThe Special Functions andTheir Approximations vol1 Academic Press New York NY USA 1969
[10] H M Srivastava and M C Daoust ldquoOn Eulerian integralsassociated with Kampe de Ferietrsquos functionrdquo Publications DeLrsquoInstitut Mathematique vol 9 no 23 pp 199ndash202 1969
[11] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Halsted Press Chichester UK
[12] E M Wright ldquoThe asymptotic expansion of the generalizedBessel functionrdquo Proceedings of London Mathematical Societyvol 38 pp 257ndash270 1935
[13] A Erdelyi Higher Transcendental Functions vol 1 McGraw-Hill New York NY USA 1953
[14] H M Srivastava K C Gupta and S P Goyal The HmdashFunctions of One and Two Variables With Applications SouthAsian Publishers New Delhi India 1982
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Engineering Mathematics 5
poles as follows
119904119903=
119889ℎ(119894)
119894+ 119896
119894
120575ℎ(119894)
119894
(ℎ119894= 1 119898
119894 119896
119894= 0 1 2 ) for 119894 = 1 119903
(19)
The behaviour of the function I[1199111119911119903
] is given by
I[[
[
1199111
119911119903
]
]
]
= 119874(
119903
prod
119895=1
1003816100381610038161003816119911119894
1003816100381610038161003816
120601119895
) max 10038161003816100381610038161199111
1003816100381610038161003816
1003816100381610038161003816119911119903
1003816100381610038161003816 997888rarr 0
(20)
where
120601119895= min
1le119895le119898119894
[
[
Re(119889(119894)
119895
120575(119894)
119895
)]
]
(119894 = 1 119903) (21)
On the other hand when |119911119894| rarr infin (119894 = 1 119903) the
associated function I1[
1199111119911119903
] given by (16) has the behaviour
I1
[
[
[
1199111
119911119903
]
]
]
= 119874(
119903
prod
119895=1
10038161003816100381610038161003816119911119895
10038161003816100381610038161003816
120601119895
) min 10038161003816100381610038161199111
1003816100381610038161003816
1003816100381610038161003816119911119903
1003816100381610038161003816 997888rarr 0
(22)
where
120601119895= max
1le119895le119899119894
[
[
Re(1 minus 119888
(119894)
119895
120574(119894)
119895
)]
]
(119894 = 1 119903) (23)
5 Elementary Special Cases
In this section we mention some interesting and usefulspecial cases of the I-function of ldquo119903rdquo variables
(i) If all the exponents 119860119895(119895 = 1 119901) 119861
119895(119895 =
1 119902) 119862(119894)
119895(119895 = 1 119901
119894 119894 = 1 119903) and
119863(119894)
119895(119895 = 1 119902
119894 119894 = 1 119903) in (1) are equal to
unity we obtain H-function of ldquo119903rdquo variables definedby Srivastava and Panda [7]
(ii) When 119901 = 119902 = 119899 = 0 (1) degenerates into the productof 119903mutually independent I- functions of one variableintroduced by Rathie [1]
(iii) When 119901 = 119902 = 119899 = 0 and 119903 = 1 (1) reduces to theI-function defined by Rathie [1]
(iv) When 119899 = 119901 119898119894= 1 119899
119894= 119901
119894 119894 = 1 119903 and 119860
119895=
119861119895= 119862
119895= 119863
119895= 1 and (119889
(119894)
119895 120575
(119894)
119895 119863
(119894)
119895) is replaced by
(0 1 1) (119889(119894)119895 120575
(119894)
119895 119863
(119894)
119895) (1) reduces to the generalized
Lauricella function [10]
I011990111199011111990111990311990111990211990111199021+1119901
119903119902119903+1
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 1)
1119901 (119888
(1)
119895 120574
(1)
119895 1)
11199011
(119888(119903)
119895 120574
(119903)
119895 1)
1119901119903
(119887119895 120573
(1)
119895 120573
(1)
119895 120573
(119903)
119895 1)
1119902 (0 1 1) (119889
(1)
119895 120575
(1)
119895 1)
11199021
(0 1 1) (119889(119903)
119895 120575
(119903)
119895 1)
1119902119903
]
]
]
=
prod119901
119895=1Γ (1 minus 119886
119895)prod
1199011
119895=1Γ (1 minus 119888
(1)
119895) sdot sdot sdotprod
119901119903
119895=1Γ (1 minus 119888
(119903)
119895)
prod119902
119895=1Γ (1 minus 119887
119895)prod
1199021
119895=1Γ (1 minus 119889
(1)
119895) sdot sdot sdotprod
119902119903
119895=1Γ (1 minus 119889
(119903)
119895)
times 1198651199011199011119901119903
1199021199021119902119903
[
[
[
minus1199111
minus119911
119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(1 minus 119886119895 120572
(1)
119895 120572
(119903)
119895)1119901
(1 minus 119888(1)
119895 120574
(1)
119895)11199011
(1 minus 119888((119903))
119895 120574
(119903)
119895)1119901119903
(1 minus 119887119895 120573
(1)
119895 120573
(119903)
119895)1119902
(1 minus 119889(1)
119895 120575
(1)
119895)11199021
(1 minus 119889(119903)
119895 120575
(119903)
119895)1119902119903
]
]
]
(24)
(v) I001119901111199011199030011990111199021+1119901
119903119902119903+1
[
[
[
minus1199111
minus119911
119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
mdash (1 minus 119888(1)
119895 120574
(1)
119895 1)
11199011
(1 minus 119888(119903)
119895 120574
(119903)
119895 1)
1119901119903
mdash (0 1 1) (1 minus 119889(1)
119895 120575
(1)
119895 1)
11199021
(0 1 1) (1 minus 119889(119903)
119895 120575
(119903)
119895 1)
1119902119903
]
]
]
=1199011
Ψ1199021
[
[
(119888(1)
119895 120574
(1)
119895)11199011
(119889(1)
119895 120575
(1)
119895)11199021
1199111]
]
times sdot sdot sdot times119901119903
Ψ119902119903
[
[
(119888(119903)
119895 120574
(119903)
119895)1119901119903
(119889(119903)
119895 120575
(119903)
119895)1119902119903
119911119903]
]
(25)
6 International Journal of Engineering Mathematics
where the functions119901119894
Ψ119902119894
119894 = 1 119903 are Wrightrsquos general-ized hypergeometric functions [11]
(vi) I001010000202
[
[
[
1199111
119911119903
10038161003816100381610038161003816100381610038161003816
mdash mdash mdashmdash (0 1 1) (minus120583
1 120572
1 1) (0 1 1) (minus120583
119903 120572
119903 1)
]
]
]
=
119903
prod
119894=1
119869120572119894
120583119894
(119911119894) (26)
where the functions 119869120572119894
120583119894
(119911119894) are Wrightrsquos generalized Bessel
functions [12]
(vii) I001212002222
[
[
[
minus1199111
minus119911
119903
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
mdash (1 1 1) (1-1205721 1 120583
1) (1 1 1) (1-120572
119903 1 120583
119903)
mdash (0 1 1) (-1205721 1 120583
1) (0 1 1) (-120572
119903 1 120583
119903)
]
]
]
=
119903
prod
119894=1
Φ(119911119894 120583
119894 120572
119894) (27)
where Φ(119911119894 120583
119894 120572
119894) 119894 = 1 119903 are the generalized Riemann
zeta functions [13 page 27 111 (1)] which are the generaliza-tions of Hurwitz zeta functions and Riemann zeta functions[13 page 24 110 (1) and 112 (1)]
(viii) I001212002222
[
[
[
minus1199111
minus119911
119903
10038161003816100381610038161003816100381610038161003816100381610038161003816
mdash (1 1 1) (1 1 1205831) (1 1 1) (1 1 120583
119903)
mdash (0 1 1) (0 1 1205831) (0 1 1) (0 1 120583
119903)
]
]
]
=
119903
prod
119894=1
119865 (119911119894 120583
119894) (28)
where 119865(119911119894 120583
119894) are the polylogarithms of order 120583
119894 For 120583
119894= 2
119894 = 1 119903 theRHS of (28) reduces to the product of Eulerrsquosdilogarithm [13 page 31 1111 equation (2)]
6 Elementary Properties andTransformation Formulas
Theproperties given below are immediate consequence of thedefinition (1) and hence they are given here without proof
(i) I001198981 119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895119861119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I001198991 1198981119899119903 11989811990311990211990111990211199011119902119903119901119903
[
[
[
119911minus1
1
119911minus1
119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(1 minus 119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(1 minus 119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(1 minus 119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
(1 minus 119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(1 minus 119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(1 minus 119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
]
]
]
(29)
(ii) 1199111198961
1sdot sdot sdot 119911
119896119903
119903I [119911
1sdot sdot sdot 119911
119903]
= I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
[
[
1199111
119911119903
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895+
119903
sum
119894=1
119896119894120572(119894)
119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)
1119901
(119888(1)
119895+119896
1120574(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895+119896
119903120574(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895+
119903
sum
119894=1
119896119894120573(119894)
119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)
1119902
(119889(1)
119895+ 119896
1120575(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895+ 119896
119903120575(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
]
]
(30)
for 119896119894gt 0 119894 = 1 119903
International Journal of Engineering Mathematics 7
(iii) 1
1198961
sdot sdot sdot
1
119896119903
I [1199111 119911
119903]
= I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
[
1199111198961
1
119911119896119903
119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 119896
1120572(1)
119895 119896
119903120572(119903)
119895 119860
119895)1119901
(119888(1)
119895 119896
1120574(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 119896
119903120574(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 119896
1120573(1)
119895 119896
119903120573(119903)
119895 119861
119895)1119901
(119889(1)
119895 119896
1120575(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 119896
119903120575(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
]
(31)
where 119896119894gt 0 119894 = 1 119903
(iv) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886 120572 0 0 119860) (119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I0119899minus111989811198991+1119898119903 1198991199031199011199021199011+11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 )
2119901 (119886 120572 119860) (119888
(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(32)
where 119901 ge 119899 ge 1
(v) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119886 120572 0 0 119860) (119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I011989911989811198991 119898119903 119899119903119901minus1119902119901
1+11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119886 120572 119860) (119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(33)
where 119901 minus 1 ge 119899 ge 0
(vi) I011989911989811198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119887 120573 0 0 119861) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I011989911989811198991 119898119903 119899119903119901119902minus1119901
11199021+1119901
119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119887 120573 119861) (119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(34)
where 119902 minus 1 ge 0
8 International Journal of Engineering Mathematics
(vii) I01198991198981 1198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886 0 0 119860) (119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= Γ119860(1 minus 119886) times I0119899minus11198981 1198991119898119903119899119903
119901minus111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(35)
where 119901 ge 119899 ge 1R(1 minus 119886) gt 0
(viii) I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119886 0 0 119860) (119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119860(119886)
times I011989911989811198991119898119903119899119903119901minus1119902119901
11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(36)
where 119901 minus 1 ge 119899 ge 0R(119886) gt 0
(ix) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119887 0 0 119861) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119861(1 minus 119887)
times I011989911989811198991 119898119903 119899119903119901119902minus1119901
11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(37)
where 119902 minus 1 ge 0R(1 minus 119887) gt 0
(x) I011989911989811198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888 0 119862) (119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= Γ119862(1 minus 119888) times I011989911989811198991minus1119898119903 119899119903
1199011199021199011minus11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(38)
where 1199011ge 119899
1ge 1R(1 minus 119888) gt 0
International Journal of Engineering Mathematics 9
(xi) I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888 0 119862) (119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119862(119888)
times I011989911989811198991 119898119903 1198991199031199011199021199011minus11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(39)
where 1199011minus 1 ge 119899
1ge 0R(119888) gt 0
(xii) I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889 0 119863) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= Γ119863(119889) times I01198991198981minus11198991 119898119903119899119903
11990111990211990111199021minus1119901
119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(40)
where 1199021ge 119898
1ge 1R(119889) gt 0
(xiii) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889 0 119863) (119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119863(1 minus 119889)
times I01198991198981119899111989811990311989911990311990111990211990111199021minus1119901
119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(41)
provided that 1199021minus 1 ge 119898
1ge 0R(1 minus 119889) ge 0
(xiv) I01198991198981 1198991 119898119903 1198991199031199011199021199011 1199021 119901119903 119902119903
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902minus1
(1198861 120572(1)
1 120572
(119903)
1 1198601) (119889
1
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119888(1)
1 120574
(1)
1 119862
(1)
1) (119889
(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
(119888(119903)
1 120574
(119903)
1 119862
(119903)
1)
]]]
]
= I0119899minus11198981 1198991minus1119898119903 119899119903minus1119901minus1119902minus11199011minus11199021minus1119901119903minus1119902119903minus1
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902minus1
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
]]]
]
(42)
provided that 119901 ge 119899 ge 1 119901119894ge 119899
119894ge 1 119894 = 1 119903 and 119902 ge 1
119902119894ge 119898
119894+ 1 119894 = 1 119903
10 International Journal of Engineering Mathematics
(xv) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119889(1)
1 120575
(1)
1 119863
(1)
1) (119888
(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119889(119903)
1 120575
(119903)
1 119863
(119903)
1)
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I01198991198981minus11198991 119898119903minus11198991199031199011199021199011minus11199021minus1119901
119903minus1119902119903minus1
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
(43)
provided that 119901 ge 119899 + 1 119902 ge 1 119901119894ge 119899
119894+ 1 and 119902
119894ge
119898119894ge 1 119894 = 1 119903
(xvi) I011989911989811198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119888(1)
1199011
120574(1)
1199011
119862(1)
1199011
) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119888(119903)
119901119903
120574(119903)
119901119903
119862(119903)
119901119903
) (119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
= I0119899 1198981minus11198991 119898
119903minus1119899119903
119901119902 1199011minus11199021minus1 119901
119903minus1119902119903minus1
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
(44)
provided that 119901119894ge 119899
119894+ 1 119902
119894ge 119898
119894 119894 = 1 119903
(xvii) I01198991198981 1198991 119898119903 1198991199031199011199021199011 1199021 119901119903 119902119903
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119889(1)
1199021 120575
(1)
1199021 119863
(1)
1199021) (119888
(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119889(119903)
119902119903 120575
(119903)
119902119903 119863
(119903)
119902119903) (119888
(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]]]
]
= I01198991198981 1198991minus1119898119903 119899119903minus11199011199021199011minus11199021minus1119901119903minus1119902119903minus1
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
]]]
]
(45)
where 119901 ge 119899 119902 ge 1 119901119894ge 119899
119894ge 1 119902
119894minus 1 ge 119898
119894 119894 = 1 119903
7 Special Cases
When 119903 = 2 and all the exponents 119860119895(119895 = 1 119901)
119861119895(119895 = 1 119902) 119862
(119894)
119895(119895 = 1 119901
119894 119894 = 1 119903) and
119863(119894)
119895(1 119902
119894 119894 = 1 119903) the I-function of ldquo119903rdquo variables
reduces to H-function of two variables and therefore weobtain the corresponding results in H-function of two vari-ables [14]
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgment
The authors are immensely grateful to the worthy referee forsome useful and valuable suggestions for the improvement ofthis paper which led to a better presentation
International Journal of Engineering Mathematics 11
References
[1] A K Rathie ldquoA new generalization of generalized hypergeo-metric functionsrdquo Le Matematiche vol 52 no 2 pp 297ndash3101997
[2] I S Ansari F Yilmaz M S Alouni and O Kucur ldquoNewresults on the sum of Gamma random variates with applicationto the performance of wireless communication systems overnakagami-m fading channelsrdquo httparxiv-web3librarycornelleduabs12022576v4
[3] I S Ansari andYilmaz ldquoOn the sumof squared n-Randomvari-ates with application to the performance of wireless communi-cation systemsrdquo httparxiv-web3librarycornelleduabs12100100v1
[4] X Minghua W Yik-Chung and A Sonia ldquoExact outage prob-ability of dual-hop CSI-assisted AF relaying over nakagami-mfading channelsrdquo IEEE Transactions on Signal Processing vol60 no 10 pp 5578ndash5583 2012
[5] K ShanthaKumari T M Vasudevan Nambisan and AK Rathie ldquoA study of the I-function of two variablesrdquohttparxivorgabs12126717
[6] P K Mittal and K C Gupta ldquoAn integral involving generalizedfunction of two variablesrdquo Proceedings of the Indian Academy ofSciences A vol 75 no 3 pp 117ndash123 1972
[7] HM Srivastava and R Panda ldquoSome bilateral generating func-tions for a class of generalkized hypergeometric polynomialsrdquoJournal Fur die Reine und Angewandte Mathematik vol 17 no288 pp 265ndash274 1976
[8] B L J Braaksma ldquoAsymptotic expansions and analytic continu-ations for a class of Barnesintegralsrdquo Compositio Mathematicahvol 15 pp 239ndash341 1964
[9] Y L LukeThe Special Functions andTheir Approximations vol1 Academic Press New York NY USA 1969
[10] H M Srivastava and M C Daoust ldquoOn Eulerian integralsassociated with Kampe de Ferietrsquos functionrdquo Publications DeLrsquoInstitut Mathematique vol 9 no 23 pp 199ndash202 1969
[11] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Halsted Press Chichester UK
[12] E M Wright ldquoThe asymptotic expansion of the generalizedBessel functionrdquo Proceedings of London Mathematical Societyvol 38 pp 257ndash270 1935
[13] A Erdelyi Higher Transcendental Functions vol 1 McGraw-Hill New York NY USA 1953
[14] H M Srivastava K C Gupta and S P Goyal The HmdashFunctions of One and Two Variables With Applications SouthAsian Publishers New Delhi India 1982
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 International Journal of Engineering Mathematics
where the functions119901119894
Ψ119902119894
119894 = 1 119903 are Wrightrsquos general-ized hypergeometric functions [11]
(vi) I001010000202
[
[
[
1199111
119911119903
10038161003816100381610038161003816100381610038161003816
mdash mdash mdashmdash (0 1 1) (minus120583
1 120572
1 1) (0 1 1) (minus120583
119903 120572
119903 1)
]
]
]
=
119903
prod
119894=1
119869120572119894
120583119894
(119911119894) (26)
where the functions 119869120572119894
120583119894
(119911119894) are Wrightrsquos generalized Bessel
functions [12]
(vii) I001212002222
[
[
[
minus1199111
minus119911
119903
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
mdash (1 1 1) (1-1205721 1 120583
1) (1 1 1) (1-120572
119903 1 120583
119903)
mdash (0 1 1) (-1205721 1 120583
1) (0 1 1) (-120572
119903 1 120583
119903)
]
]
]
=
119903
prod
119894=1
Φ(119911119894 120583
119894 120572
119894) (27)
where Φ(119911119894 120583
119894 120572
119894) 119894 = 1 119903 are the generalized Riemann
zeta functions [13 page 27 111 (1)] which are the generaliza-tions of Hurwitz zeta functions and Riemann zeta functions[13 page 24 110 (1) and 112 (1)]
(viii) I001212002222
[
[
[
minus1199111
minus119911
119903
10038161003816100381610038161003816100381610038161003816100381610038161003816
mdash (1 1 1) (1 1 1205831) (1 1 1) (1 1 120583
119903)
mdash (0 1 1) (0 1 1205831) (0 1 1) (0 1 120583
119903)
]
]
]
=
119903
prod
119894=1
119865 (119911119894 120583
119894) (28)
where 119865(119911119894 120583
119894) are the polylogarithms of order 120583
119894 For 120583
119894= 2
119894 = 1 119903 theRHS of (28) reduces to the product of Eulerrsquosdilogarithm [13 page 31 1111 equation (2)]
6 Elementary Properties andTransformation Formulas
Theproperties given below are immediate consequence of thedefinition (1) and hence they are given here without proof
(i) I001198981 119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895119861119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I001198991 1198981119899119903 11989811990311990211990111990211199011119902119903119901119903
[
[
[
119911minus1
1
119911minus1
119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(1 minus 119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(1 minus 119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(1 minus 119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
(1 minus 119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(1 minus 119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(1 minus 119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
]
]
]
(29)
(ii) 1199111198961
1sdot sdot sdot 119911
119896119903
119903I [119911
1sdot sdot sdot 119911
119903]
= I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
[
[
1199111
119911119903
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895+
119903
sum
119894=1
119896119894120572(119894)
119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)
1119901
(119888(1)
119895+119896
1120574(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895+119896
119903120574(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895+
119903
sum
119894=1
119896119894120573(119894)
119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)
1119902
(119889(1)
119895+ 119896
1120575(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895+ 119896
119903120575(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
]
]
(30)
for 119896119894gt 0 119894 = 1 119903
International Journal of Engineering Mathematics 7
(iii) 1
1198961
sdot sdot sdot
1
119896119903
I [1199111 119911
119903]
= I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
[
1199111198961
1
119911119896119903
119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 119896
1120572(1)
119895 119896
119903120572(119903)
119895 119860
119895)1119901
(119888(1)
119895 119896
1120574(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 119896
119903120574(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 119896
1120573(1)
119895 119896
119903120573(119903)
119895 119861
119895)1119901
(119889(1)
119895 119896
1120575(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 119896
119903120575(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
]
(31)
where 119896119894gt 0 119894 = 1 119903
(iv) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886 120572 0 0 119860) (119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I0119899minus111989811198991+1119898119903 1198991199031199011199021199011+11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 )
2119901 (119886 120572 119860) (119888
(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(32)
where 119901 ge 119899 ge 1
(v) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119886 120572 0 0 119860) (119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I011989911989811198991 119898119903 119899119903119901minus1119902119901
1+11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119886 120572 119860) (119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(33)
where 119901 minus 1 ge 119899 ge 0
(vi) I011989911989811198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119887 120573 0 0 119861) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I011989911989811198991 119898119903 119899119903119901119902minus1119901
11199021+1119901
119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119887 120573 119861) (119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(34)
where 119902 minus 1 ge 0
8 International Journal of Engineering Mathematics
(vii) I01198991198981 1198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886 0 0 119860) (119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= Γ119860(1 minus 119886) times I0119899minus11198981 1198991119898119903119899119903
119901minus111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(35)
where 119901 ge 119899 ge 1R(1 minus 119886) gt 0
(viii) I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119886 0 0 119860) (119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119860(119886)
times I011989911989811198991119898119903119899119903119901minus1119902119901
11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(36)
where 119901 minus 1 ge 119899 ge 0R(119886) gt 0
(ix) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119887 0 0 119861) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119861(1 minus 119887)
times I011989911989811198991 119898119903 119899119903119901119902minus1119901
11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(37)
where 119902 minus 1 ge 0R(1 minus 119887) gt 0
(x) I011989911989811198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888 0 119862) (119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= Γ119862(1 minus 119888) times I011989911989811198991minus1119898119903 119899119903
1199011199021199011minus11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(38)
where 1199011ge 119899
1ge 1R(1 minus 119888) gt 0
International Journal of Engineering Mathematics 9
(xi) I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888 0 119862) (119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119862(119888)
times I011989911989811198991 119898119903 1198991199031199011199021199011minus11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(39)
where 1199011minus 1 ge 119899
1ge 0R(119888) gt 0
(xii) I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889 0 119863) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= Γ119863(119889) times I01198991198981minus11198991 119898119903119899119903
11990111990211990111199021minus1119901
119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(40)
where 1199021ge 119898
1ge 1R(119889) gt 0
(xiii) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889 0 119863) (119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119863(1 minus 119889)
times I01198991198981119899111989811990311989911990311990111990211990111199021minus1119901
119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(41)
provided that 1199021minus 1 ge 119898
1ge 0R(1 minus 119889) ge 0
(xiv) I01198991198981 1198991 119898119903 1198991199031199011199021199011 1199021 119901119903 119902119903
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902minus1
(1198861 120572(1)
1 120572
(119903)
1 1198601) (119889
1
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119888(1)
1 120574
(1)
1 119862
(1)
1) (119889
(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
(119888(119903)
1 120574
(119903)
1 119862
(119903)
1)
]]]
]
= I0119899minus11198981 1198991minus1119898119903 119899119903minus1119901minus1119902minus11199011minus11199021minus1119901119903minus1119902119903minus1
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902minus1
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
]]]
]
(42)
provided that 119901 ge 119899 ge 1 119901119894ge 119899
119894ge 1 119894 = 1 119903 and 119902 ge 1
119902119894ge 119898
119894+ 1 119894 = 1 119903
10 International Journal of Engineering Mathematics
(xv) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119889(1)
1 120575
(1)
1 119863
(1)
1) (119888
(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119889(119903)
1 120575
(119903)
1 119863
(119903)
1)
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I01198991198981minus11198991 119898119903minus11198991199031199011199021199011minus11199021minus1119901
119903minus1119902119903minus1
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
(43)
provided that 119901 ge 119899 + 1 119902 ge 1 119901119894ge 119899
119894+ 1 and 119902
119894ge
119898119894ge 1 119894 = 1 119903
(xvi) I011989911989811198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119888(1)
1199011
120574(1)
1199011
119862(1)
1199011
) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119888(119903)
119901119903
120574(119903)
119901119903
119862(119903)
119901119903
) (119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
= I0119899 1198981minus11198991 119898
119903minus1119899119903
119901119902 1199011minus11199021minus1 119901
119903minus1119902119903minus1
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
(44)
provided that 119901119894ge 119899
119894+ 1 119902
119894ge 119898
119894 119894 = 1 119903
(xvii) I01198991198981 1198991 119898119903 1198991199031199011199021199011 1199021 119901119903 119902119903
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119889(1)
1199021 120575
(1)
1199021 119863
(1)
1199021) (119888
(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119889(119903)
119902119903 120575
(119903)
119902119903 119863
(119903)
119902119903) (119888
(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]]]
]
= I01198991198981 1198991minus1119898119903 119899119903minus11199011199021199011minus11199021minus1119901119903minus1119902119903minus1
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
]]]
]
(45)
where 119901 ge 119899 119902 ge 1 119901119894ge 119899
119894ge 1 119902
119894minus 1 ge 119898
119894 119894 = 1 119903
7 Special Cases
When 119903 = 2 and all the exponents 119860119895(119895 = 1 119901)
119861119895(119895 = 1 119902) 119862
(119894)
119895(119895 = 1 119901
119894 119894 = 1 119903) and
119863(119894)
119895(1 119902
119894 119894 = 1 119903) the I-function of ldquo119903rdquo variables
reduces to H-function of two variables and therefore weobtain the corresponding results in H-function of two vari-ables [14]
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgment
The authors are immensely grateful to the worthy referee forsome useful and valuable suggestions for the improvement ofthis paper which led to a better presentation
International Journal of Engineering Mathematics 11
References
[1] A K Rathie ldquoA new generalization of generalized hypergeo-metric functionsrdquo Le Matematiche vol 52 no 2 pp 297ndash3101997
[2] I S Ansari F Yilmaz M S Alouni and O Kucur ldquoNewresults on the sum of Gamma random variates with applicationto the performance of wireless communication systems overnakagami-m fading channelsrdquo httparxiv-web3librarycornelleduabs12022576v4
[3] I S Ansari andYilmaz ldquoOn the sumof squared n-Randomvari-ates with application to the performance of wireless communi-cation systemsrdquo httparxiv-web3librarycornelleduabs12100100v1
[4] X Minghua W Yik-Chung and A Sonia ldquoExact outage prob-ability of dual-hop CSI-assisted AF relaying over nakagami-mfading channelsrdquo IEEE Transactions on Signal Processing vol60 no 10 pp 5578ndash5583 2012
[5] K ShanthaKumari T M Vasudevan Nambisan and AK Rathie ldquoA study of the I-function of two variablesrdquohttparxivorgabs12126717
[6] P K Mittal and K C Gupta ldquoAn integral involving generalizedfunction of two variablesrdquo Proceedings of the Indian Academy ofSciences A vol 75 no 3 pp 117ndash123 1972
[7] HM Srivastava and R Panda ldquoSome bilateral generating func-tions for a class of generalkized hypergeometric polynomialsrdquoJournal Fur die Reine und Angewandte Mathematik vol 17 no288 pp 265ndash274 1976
[8] B L J Braaksma ldquoAsymptotic expansions and analytic continu-ations for a class of Barnesintegralsrdquo Compositio Mathematicahvol 15 pp 239ndash341 1964
[9] Y L LukeThe Special Functions andTheir Approximations vol1 Academic Press New York NY USA 1969
[10] H M Srivastava and M C Daoust ldquoOn Eulerian integralsassociated with Kampe de Ferietrsquos functionrdquo Publications DeLrsquoInstitut Mathematique vol 9 no 23 pp 199ndash202 1969
[11] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Halsted Press Chichester UK
[12] E M Wright ldquoThe asymptotic expansion of the generalizedBessel functionrdquo Proceedings of London Mathematical Societyvol 38 pp 257ndash270 1935
[13] A Erdelyi Higher Transcendental Functions vol 1 McGraw-Hill New York NY USA 1953
[14] H M Srivastava K C Gupta and S P Goyal The HmdashFunctions of One and Two Variables With Applications SouthAsian Publishers New Delhi India 1982
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Engineering Mathematics 7
(iii) 1
1198961
sdot sdot sdot
1
119896119903
I [1199111 119911
119903]
= I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
[
1199111198961
1
119911119896119903
119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 119896
1120572(1)
119895 119896
119903120572(119903)
119895 119860
119895)1119901
(119888(1)
119895 119896
1120574(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 119896
119903120574(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 119896
1120573(1)
119895 119896
119903120573(119903)
119895 119861
119895)1119901
(119889(1)
119895 119896
1120575(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 119896
119903120575(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
]
(31)
where 119896119894gt 0 119894 = 1 119903
(iv) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886 120572 0 0 119860) (119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I0119899minus111989811198991+1119898119903 1198991199031199011199021199011+11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 )
2119901 (119886 120572 119860) (119888
(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(32)
where 119901 ge 119899 ge 1
(v) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119886 120572 0 0 119860) (119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I011989911989811198991 119898119903 119899119903119901minus1119902119901
1+11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119886 120572 119860) (119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(33)
where 119901 minus 1 ge 119899 ge 0
(vi) I011989911989811198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119887 120573 0 0 119861) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I011989911989811198991 119898119903 119899119903119901119902minus1119901
11199021+1119901
119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119887 120573 119861) (119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(34)
where 119902 minus 1 ge 0
8 International Journal of Engineering Mathematics
(vii) I01198991198981 1198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886 0 0 119860) (119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= Γ119860(1 minus 119886) times I0119899minus11198981 1198991119898119903119899119903
119901minus111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(35)
where 119901 ge 119899 ge 1R(1 minus 119886) gt 0
(viii) I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119886 0 0 119860) (119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119860(119886)
times I011989911989811198991119898119903119899119903119901minus1119902119901
11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(36)
where 119901 minus 1 ge 119899 ge 0R(119886) gt 0
(ix) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119887 0 0 119861) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119861(1 minus 119887)
times I011989911989811198991 119898119903 119899119903119901119902minus1119901
11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(37)
where 119902 minus 1 ge 0R(1 minus 119887) gt 0
(x) I011989911989811198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888 0 119862) (119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= Γ119862(1 minus 119888) times I011989911989811198991minus1119898119903 119899119903
1199011199021199011minus11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(38)
where 1199011ge 119899
1ge 1R(1 minus 119888) gt 0
International Journal of Engineering Mathematics 9
(xi) I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888 0 119862) (119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119862(119888)
times I011989911989811198991 119898119903 1198991199031199011199021199011minus11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(39)
where 1199011minus 1 ge 119899
1ge 0R(119888) gt 0
(xii) I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889 0 119863) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= Γ119863(119889) times I01198991198981minus11198991 119898119903119899119903
11990111990211990111199021minus1119901
119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(40)
where 1199021ge 119898
1ge 1R(119889) gt 0
(xiii) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889 0 119863) (119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119863(1 minus 119889)
times I01198991198981119899111989811990311989911990311990111990211990111199021minus1119901
119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(41)
provided that 1199021minus 1 ge 119898
1ge 0R(1 minus 119889) ge 0
(xiv) I01198991198981 1198991 119898119903 1198991199031199011199021199011 1199021 119901119903 119902119903
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902minus1
(1198861 120572(1)
1 120572
(119903)
1 1198601) (119889
1
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119888(1)
1 120574
(1)
1 119862
(1)
1) (119889
(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
(119888(119903)
1 120574
(119903)
1 119862
(119903)
1)
]]]
]
= I0119899minus11198981 1198991minus1119898119903 119899119903minus1119901minus1119902minus11199011minus11199021minus1119901119903minus1119902119903minus1
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902minus1
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
]]]
]
(42)
provided that 119901 ge 119899 ge 1 119901119894ge 119899
119894ge 1 119894 = 1 119903 and 119902 ge 1
119902119894ge 119898
119894+ 1 119894 = 1 119903
10 International Journal of Engineering Mathematics
(xv) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119889(1)
1 120575
(1)
1 119863
(1)
1) (119888
(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119889(119903)
1 120575
(119903)
1 119863
(119903)
1)
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I01198991198981minus11198991 119898119903minus11198991199031199011199021199011minus11199021minus1119901
119903minus1119902119903minus1
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
(43)
provided that 119901 ge 119899 + 1 119902 ge 1 119901119894ge 119899
119894+ 1 and 119902
119894ge
119898119894ge 1 119894 = 1 119903
(xvi) I011989911989811198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119888(1)
1199011
120574(1)
1199011
119862(1)
1199011
) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119888(119903)
119901119903
120574(119903)
119901119903
119862(119903)
119901119903
) (119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
= I0119899 1198981minus11198991 119898
119903minus1119899119903
119901119902 1199011minus11199021minus1 119901
119903minus1119902119903minus1
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
(44)
provided that 119901119894ge 119899
119894+ 1 119902
119894ge 119898
119894 119894 = 1 119903
(xvii) I01198991198981 1198991 119898119903 1198991199031199011199021199011 1199021 119901119903 119902119903
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119889(1)
1199021 120575
(1)
1199021 119863
(1)
1199021) (119888
(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119889(119903)
119902119903 120575
(119903)
119902119903 119863
(119903)
119902119903) (119888
(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]]]
]
= I01198991198981 1198991minus1119898119903 119899119903minus11199011199021199011minus11199021minus1119901119903minus1119902119903minus1
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
]]]
]
(45)
where 119901 ge 119899 119902 ge 1 119901119894ge 119899
119894ge 1 119902
119894minus 1 ge 119898
119894 119894 = 1 119903
7 Special Cases
When 119903 = 2 and all the exponents 119860119895(119895 = 1 119901)
119861119895(119895 = 1 119902) 119862
(119894)
119895(119895 = 1 119901
119894 119894 = 1 119903) and
119863(119894)
119895(1 119902
119894 119894 = 1 119903) the I-function of ldquo119903rdquo variables
reduces to H-function of two variables and therefore weobtain the corresponding results in H-function of two vari-ables [14]
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgment
The authors are immensely grateful to the worthy referee forsome useful and valuable suggestions for the improvement ofthis paper which led to a better presentation
International Journal of Engineering Mathematics 11
References
[1] A K Rathie ldquoA new generalization of generalized hypergeo-metric functionsrdquo Le Matematiche vol 52 no 2 pp 297ndash3101997
[2] I S Ansari F Yilmaz M S Alouni and O Kucur ldquoNewresults on the sum of Gamma random variates with applicationto the performance of wireless communication systems overnakagami-m fading channelsrdquo httparxiv-web3librarycornelleduabs12022576v4
[3] I S Ansari andYilmaz ldquoOn the sumof squared n-Randomvari-ates with application to the performance of wireless communi-cation systemsrdquo httparxiv-web3librarycornelleduabs12100100v1
[4] X Minghua W Yik-Chung and A Sonia ldquoExact outage prob-ability of dual-hop CSI-assisted AF relaying over nakagami-mfading channelsrdquo IEEE Transactions on Signal Processing vol60 no 10 pp 5578ndash5583 2012
[5] K ShanthaKumari T M Vasudevan Nambisan and AK Rathie ldquoA study of the I-function of two variablesrdquohttparxivorgabs12126717
[6] P K Mittal and K C Gupta ldquoAn integral involving generalizedfunction of two variablesrdquo Proceedings of the Indian Academy ofSciences A vol 75 no 3 pp 117ndash123 1972
[7] HM Srivastava and R Panda ldquoSome bilateral generating func-tions for a class of generalkized hypergeometric polynomialsrdquoJournal Fur die Reine und Angewandte Mathematik vol 17 no288 pp 265ndash274 1976
[8] B L J Braaksma ldquoAsymptotic expansions and analytic continu-ations for a class of Barnesintegralsrdquo Compositio Mathematicahvol 15 pp 239ndash341 1964
[9] Y L LukeThe Special Functions andTheir Approximations vol1 Academic Press New York NY USA 1969
[10] H M Srivastava and M C Daoust ldquoOn Eulerian integralsassociated with Kampe de Ferietrsquos functionrdquo Publications DeLrsquoInstitut Mathematique vol 9 no 23 pp 199ndash202 1969
[11] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Halsted Press Chichester UK
[12] E M Wright ldquoThe asymptotic expansion of the generalizedBessel functionrdquo Proceedings of London Mathematical Societyvol 38 pp 257ndash270 1935
[13] A Erdelyi Higher Transcendental Functions vol 1 McGraw-Hill New York NY USA 1953
[14] H M Srivastava K C Gupta and S P Goyal The HmdashFunctions of One and Two Variables With Applications SouthAsian Publishers New Delhi India 1982
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 International Journal of Engineering Mathematics
(vii) I01198991198981 1198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886 0 0 119860) (119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= Γ119860(1 minus 119886) times I0119899minus11198981 1198991119898119903119899119903
119901minus111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(35)
where 119901 ge 119899 ge 1R(1 minus 119886) gt 0
(viii) I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119886 0 0 119860) (119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119860(119886)
times I011989911989811198991119898119903119899119903119901minus1119902119901
11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901minus1
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(36)
where 119901 minus 1 ge 119899 ge 0R(119886) gt 0
(ix) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119887 0 0 119861) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119861(1 minus 119887)
times I011989911989811198991 119898119903 119899119903119901119902minus1119901
11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902minus1
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(37)
where 119902 minus 1 ge 0R(1 minus 119887) gt 0
(x) I011989911989811198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888 0 119862) (119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= Γ119862(1 minus 119888) times I011989911989811198991minus1119898119903 119899119903
1199011199021199011minus11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(38)
where 1199011ge 119899
1ge 1R(1 minus 119888) gt 0
International Journal of Engineering Mathematics 9
(xi) I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888 0 119862) (119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119862(119888)
times I011989911989811198991 119898119903 1198991199031199011199021199011minus11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(39)
where 1199011minus 1 ge 119899
1ge 0R(119888) gt 0
(xii) I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889 0 119863) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= Γ119863(119889) times I01198991198981minus11198991 119898119903119899119903
11990111990211990111199021minus1119901
119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(40)
where 1199021ge 119898
1ge 1R(119889) gt 0
(xiii) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889 0 119863) (119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119863(1 minus 119889)
times I01198991198981119899111989811990311989911990311990111990211990111199021minus1119901
119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(41)
provided that 1199021minus 1 ge 119898
1ge 0R(1 minus 119889) ge 0
(xiv) I01198991198981 1198991 119898119903 1198991199031199011199021199011 1199021 119901119903 119902119903
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902minus1
(1198861 120572(1)
1 120572
(119903)
1 1198601) (119889
1
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119888(1)
1 120574
(1)
1 119862
(1)
1) (119889
(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
(119888(119903)
1 120574
(119903)
1 119862
(119903)
1)
]]]
]
= I0119899minus11198981 1198991minus1119898119903 119899119903minus1119901minus1119902minus11199011minus11199021minus1119901119903minus1119902119903minus1
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902minus1
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
]]]
]
(42)
provided that 119901 ge 119899 ge 1 119901119894ge 119899
119894ge 1 119894 = 1 119903 and 119902 ge 1
119902119894ge 119898
119894+ 1 119894 = 1 119903
10 International Journal of Engineering Mathematics
(xv) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119889(1)
1 120575
(1)
1 119863
(1)
1) (119888
(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119889(119903)
1 120575
(119903)
1 119863
(119903)
1)
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I01198991198981minus11198991 119898119903minus11198991199031199011199021199011minus11199021minus1119901
119903minus1119902119903minus1
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
(43)
provided that 119901 ge 119899 + 1 119902 ge 1 119901119894ge 119899
119894+ 1 and 119902
119894ge
119898119894ge 1 119894 = 1 119903
(xvi) I011989911989811198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119888(1)
1199011
120574(1)
1199011
119862(1)
1199011
) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119888(119903)
119901119903
120574(119903)
119901119903
119862(119903)
119901119903
) (119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
= I0119899 1198981minus11198991 119898
119903minus1119899119903
119901119902 1199011minus11199021minus1 119901
119903minus1119902119903minus1
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
(44)
provided that 119901119894ge 119899
119894+ 1 119902
119894ge 119898
119894 119894 = 1 119903
(xvii) I01198991198981 1198991 119898119903 1198991199031199011199021199011 1199021 119901119903 119902119903
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119889(1)
1199021 120575
(1)
1199021 119863
(1)
1199021) (119888
(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119889(119903)
119902119903 120575
(119903)
119902119903 119863
(119903)
119902119903) (119888
(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]]]
]
= I01198991198981 1198991minus1119898119903 119899119903minus11199011199021199011minus11199021minus1119901119903minus1119902119903minus1
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
]]]
]
(45)
where 119901 ge 119899 119902 ge 1 119901119894ge 119899
119894ge 1 119902
119894minus 1 ge 119898
119894 119894 = 1 119903
7 Special Cases
When 119903 = 2 and all the exponents 119860119895(119895 = 1 119901)
119861119895(119895 = 1 119902) 119862
(119894)
119895(119895 = 1 119901
119894 119894 = 1 119903) and
119863(119894)
119895(1 119902
119894 119894 = 1 119903) the I-function of ldquo119903rdquo variables
reduces to H-function of two variables and therefore weobtain the corresponding results in H-function of two vari-ables [14]
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgment
The authors are immensely grateful to the worthy referee forsome useful and valuable suggestions for the improvement ofthis paper which led to a better presentation
International Journal of Engineering Mathematics 11
References
[1] A K Rathie ldquoA new generalization of generalized hypergeo-metric functionsrdquo Le Matematiche vol 52 no 2 pp 297ndash3101997
[2] I S Ansari F Yilmaz M S Alouni and O Kucur ldquoNewresults on the sum of Gamma random variates with applicationto the performance of wireless communication systems overnakagami-m fading channelsrdquo httparxiv-web3librarycornelleduabs12022576v4
[3] I S Ansari andYilmaz ldquoOn the sumof squared n-Randomvari-ates with application to the performance of wireless communi-cation systemsrdquo httparxiv-web3librarycornelleduabs12100100v1
[4] X Minghua W Yik-Chung and A Sonia ldquoExact outage prob-ability of dual-hop CSI-assisted AF relaying over nakagami-mfading channelsrdquo IEEE Transactions on Signal Processing vol60 no 10 pp 5578ndash5583 2012
[5] K ShanthaKumari T M Vasudevan Nambisan and AK Rathie ldquoA study of the I-function of two variablesrdquohttparxivorgabs12126717
[6] P K Mittal and K C Gupta ldquoAn integral involving generalizedfunction of two variablesrdquo Proceedings of the Indian Academy ofSciences A vol 75 no 3 pp 117ndash123 1972
[7] HM Srivastava and R Panda ldquoSome bilateral generating func-tions for a class of generalkized hypergeometric polynomialsrdquoJournal Fur die Reine und Angewandte Mathematik vol 17 no288 pp 265ndash274 1976
[8] B L J Braaksma ldquoAsymptotic expansions and analytic continu-ations for a class of Barnesintegralsrdquo Compositio Mathematicahvol 15 pp 239ndash341 1964
[9] Y L LukeThe Special Functions andTheir Approximations vol1 Academic Press New York NY USA 1969
[10] H M Srivastava and M C Daoust ldquoOn Eulerian integralsassociated with Kampe de Ferietrsquos functionrdquo Publications DeLrsquoInstitut Mathematique vol 9 no 23 pp 199ndash202 1969
[11] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Halsted Press Chichester UK
[12] E M Wright ldquoThe asymptotic expansion of the generalizedBessel functionrdquo Proceedings of London Mathematical Societyvol 38 pp 257ndash270 1935
[13] A Erdelyi Higher Transcendental Functions vol 1 McGraw-Hill New York NY USA 1953
[14] H M Srivastava K C Gupta and S P Goyal The HmdashFunctions of One and Two Variables With Applications SouthAsian Publishers New Delhi India 1982
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Engineering Mathematics 9
(xi) I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888 0 119862) (119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119862(119888)
times I011989911989811198991 119898119903 1198991199031199011199021199011minus11199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(39)
where 1199011minus 1 ge 119899
1ge 0R(119888) gt 0
(xii) I011989911989811198991 119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889 0 119863) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= Γ119863(119889) times I01198991198981minus11198991 119898119903119899119903
11990111990211990111199021minus1119901
119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(40)
where 1199021ge 119898
1ge 1R(119889) gt 0
(xiii) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889 0 119863) (119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
=
1
Γ119863(1 minus 119889)
times I01198991198981119899111989811990311989911990311990111990211990111199021minus1119901
119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
(41)
provided that 1199021minus 1 ge 119898
1ge 0R(1 minus 119889) ge 0
(xiv) I01198991198981 1198991 119898119903 1198991199031199011199021199011 1199021 119901119903 119902119903
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902minus1
(1198861 120572(1)
1 120572
(119903)
1 1198601) (119889
1
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119888(1)
1 120574
(1)
1 119862
(1)
1) (119889
(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
(119888(119903)
1 120574
(119903)
1 119862
(119903)
1)
]]]
]
= I0119899minus11198981 1198991minus1119898119903 119899119903minus1119901minus1119902minus11199011minus11199021minus1119901119903minus1119902119903minus1
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
2119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902minus1
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
]]]
]
(42)
provided that 119901 ge 119899 ge 1 119901119894ge 119899
119894ge 1 119894 = 1 119903 and 119902 ge 1
119902119894ge 119898
119894+ 1 119894 = 1 119903
10 International Journal of Engineering Mathematics
(xv) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119889(1)
1 120575
(1)
1 119863
(1)
1) (119888
(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119889(119903)
1 120575
(119903)
1 119863
(119903)
1)
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I01198991198981minus11198991 119898119903minus11198991199031199011199021199011minus11199021minus1119901
119903minus1119902119903minus1
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
(43)
provided that 119901 ge 119899 + 1 119902 ge 1 119901119894ge 119899
119894+ 1 and 119902
119894ge
119898119894ge 1 119894 = 1 119903
(xvi) I011989911989811198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119888(1)
1199011
120574(1)
1199011
119862(1)
1199011
) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119888(119903)
119901119903
120574(119903)
119901119903
119862(119903)
119901119903
) (119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
= I0119899 1198981minus11198991 119898
119903minus1119899119903
119901119902 1199011minus11199021minus1 119901
119903minus1119902119903minus1
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
(44)
provided that 119901119894ge 119899
119894+ 1 119902
119894ge 119898
119894 119894 = 1 119903
(xvii) I01198991198981 1198991 119898119903 1198991199031199011199021199011 1199021 119901119903 119902119903
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119889(1)
1199021 120575
(1)
1199021 119863
(1)
1199021) (119888
(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119889(119903)
119902119903 120575
(119903)
119902119903 119863
(119903)
119902119903) (119888
(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]]]
]
= I01198991198981 1198991minus1119898119903 119899119903minus11199011199021199011minus11199021minus1119901119903minus1119902119903minus1
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
]]]
]
(45)
where 119901 ge 119899 119902 ge 1 119901119894ge 119899
119894ge 1 119902
119894minus 1 ge 119898
119894 119894 = 1 119903
7 Special Cases
When 119903 = 2 and all the exponents 119860119895(119895 = 1 119901)
119861119895(119895 = 1 119902) 119862
(119894)
119895(119895 = 1 119901
119894 119894 = 1 119903) and
119863(119894)
119895(1 119902
119894 119894 = 1 119903) the I-function of ldquo119903rdquo variables
reduces to H-function of two variables and therefore weobtain the corresponding results in H-function of two vari-ables [14]
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgment
The authors are immensely grateful to the worthy referee forsome useful and valuable suggestions for the improvement ofthis paper which led to a better presentation
International Journal of Engineering Mathematics 11
References
[1] A K Rathie ldquoA new generalization of generalized hypergeo-metric functionsrdquo Le Matematiche vol 52 no 2 pp 297ndash3101997
[2] I S Ansari F Yilmaz M S Alouni and O Kucur ldquoNewresults on the sum of Gamma random variates with applicationto the performance of wireless communication systems overnakagami-m fading channelsrdquo httparxiv-web3librarycornelleduabs12022576v4
[3] I S Ansari andYilmaz ldquoOn the sumof squared n-Randomvari-ates with application to the performance of wireless communi-cation systemsrdquo httparxiv-web3librarycornelleduabs12100100v1
[4] X Minghua W Yik-Chung and A Sonia ldquoExact outage prob-ability of dual-hop CSI-assisted AF relaying over nakagami-mfading channelsrdquo IEEE Transactions on Signal Processing vol60 no 10 pp 5578ndash5583 2012
[5] K ShanthaKumari T M Vasudevan Nambisan and AK Rathie ldquoA study of the I-function of two variablesrdquohttparxivorgabs12126717
[6] P K Mittal and K C Gupta ldquoAn integral involving generalizedfunction of two variablesrdquo Proceedings of the Indian Academy ofSciences A vol 75 no 3 pp 117ndash123 1972
[7] HM Srivastava and R Panda ldquoSome bilateral generating func-tions for a class of generalkized hypergeometric polynomialsrdquoJournal Fur die Reine und Angewandte Mathematik vol 17 no288 pp 265ndash274 1976
[8] B L J Braaksma ldquoAsymptotic expansions and analytic continu-ations for a class of Barnesintegralsrdquo Compositio Mathematicahvol 15 pp 239ndash341 1964
[9] Y L LukeThe Special Functions andTheir Approximations vol1 Academic Press New York NY USA 1969
[10] H M Srivastava and M C Daoust ldquoOn Eulerian integralsassociated with Kampe de Ferietrsquos functionrdquo Publications DeLrsquoInstitut Mathematique vol 9 no 23 pp 199ndash202 1969
[11] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Halsted Press Chichester UK
[12] E M Wright ldquoThe asymptotic expansion of the generalizedBessel functionrdquo Proceedings of London Mathematical Societyvol 38 pp 257ndash270 1935
[13] A Erdelyi Higher Transcendental Functions vol 1 McGraw-Hill New York NY USA 1953
[14] H M Srivastava K C Gupta and S P Goyal The HmdashFunctions of One and Two Variables With Applications SouthAsian Publishers New Delhi India 1982
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 International Journal of Engineering Mathematics
(xv) I01198991198981119899111989811990311989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119889(1)
1 120575
(1)
1 119863
(1)
1) (119888
(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119889(119903)
1 120575
(119903)
1 119863
(119903)
1)
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]
]
]
= I01198991198981minus11198991 119898119903minus11198991199031199011199021199011minus11199021minus1119901
119903minus1119902119903minus1
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
(43)
provided that 119901 ge 119899 + 1 119902 ge 1 119901119894ge 119899
119894+ 1 and 119902
119894ge
119898119894ge 1 119894 = 1 119903
(xvi) I011989911989811198991119898119903 11989911990311990111990211990111199021119901119903119902119903
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119888(1)
1199011
120574(1)
1199011
119862(1)
1199011
) (119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119888(119903)
119901119903
120574(119903)
119901119903
119862(119903)
119901119903
) (119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
= I0119899 1198981minus11198991 119898
119903minus1119899119903
119901119902 1199011minus11199021minus1 119901
119903minus1119902119903minus1
[
[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572
(1)
119895 120572
(119903)
119895 119860
119895)1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)11199011minus1
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)1119901119903minus1
(119887119895 120573
(1)
119895 120573
(119903)
119895 119861
119895)1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)21199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)2119902119903
]
]
]
(44)
provided that 119901119894ge 119899
119894+ 1 119902
119894ge 119898
119894 119894 = 1 119903
(xvii) I01198991198981 1198991 119898119903 1198991199031199011199021199011 1199021 119901119903 119902119903
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119889(1)
1199021 120575
(1)
1199021 119863
(1)
1199021) (119888
(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119889(119903)
119902119903 120575
(119903)
119902119903 119863
(119903)
119902119903) (119888
(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903
]]]
]
= I01198991198981 1198991minus1119898119903 119899119903minus11199011199021199011minus11199021minus1119901119903minus1119902119903minus1
[[[
[
1199111
119911119903
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119886119895 120572(1)
119895 120572
(119903)
119895 119860119895)
1119901
(119888(1)
119895 120574
(1)
119895 119862
(1)
119895)21199011
(119888(119903)
119895 120574
(119903)
119895 119862
(119903)
119895)2119901119903
(119887119895 120573(1)
119895 120573
(119903)
119895 119861119895)
1119902
(119889(1)
119895 120575
(1)
119895 119863
(1)
119895)11199021minus1
(119889(119903)
119895 120575
(119903)
119895 119863
(119903)
119895)1119902119903minus1
]]]
]
(45)
where 119901 ge 119899 119902 ge 1 119901119894ge 119899
119894ge 1 119902
119894minus 1 ge 119898
119894 119894 = 1 119903
7 Special Cases
When 119903 = 2 and all the exponents 119860119895(119895 = 1 119901)
119861119895(119895 = 1 119902) 119862
(119894)
119895(119895 = 1 119901
119894 119894 = 1 119903) and
119863(119894)
119895(1 119902
119894 119894 = 1 119903) the I-function of ldquo119903rdquo variables
reduces to H-function of two variables and therefore weobtain the corresponding results in H-function of two vari-ables [14]
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper
Acknowledgment
The authors are immensely grateful to the worthy referee forsome useful and valuable suggestions for the improvement ofthis paper which led to a better presentation
International Journal of Engineering Mathematics 11
References
[1] A K Rathie ldquoA new generalization of generalized hypergeo-metric functionsrdquo Le Matematiche vol 52 no 2 pp 297ndash3101997
[2] I S Ansari F Yilmaz M S Alouni and O Kucur ldquoNewresults on the sum of Gamma random variates with applicationto the performance of wireless communication systems overnakagami-m fading channelsrdquo httparxiv-web3librarycornelleduabs12022576v4
[3] I S Ansari andYilmaz ldquoOn the sumof squared n-Randomvari-ates with application to the performance of wireless communi-cation systemsrdquo httparxiv-web3librarycornelleduabs12100100v1
[4] X Minghua W Yik-Chung and A Sonia ldquoExact outage prob-ability of dual-hop CSI-assisted AF relaying over nakagami-mfading channelsrdquo IEEE Transactions on Signal Processing vol60 no 10 pp 5578ndash5583 2012
[5] K ShanthaKumari T M Vasudevan Nambisan and AK Rathie ldquoA study of the I-function of two variablesrdquohttparxivorgabs12126717
[6] P K Mittal and K C Gupta ldquoAn integral involving generalizedfunction of two variablesrdquo Proceedings of the Indian Academy ofSciences A vol 75 no 3 pp 117ndash123 1972
[7] HM Srivastava and R Panda ldquoSome bilateral generating func-tions for a class of generalkized hypergeometric polynomialsrdquoJournal Fur die Reine und Angewandte Mathematik vol 17 no288 pp 265ndash274 1976
[8] B L J Braaksma ldquoAsymptotic expansions and analytic continu-ations for a class of Barnesintegralsrdquo Compositio Mathematicahvol 15 pp 239ndash341 1964
[9] Y L LukeThe Special Functions andTheir Approximations vol1 Academic Press New York NY USA 1969
[10] H M Srivastava and M C Daoust ldquoOn Eulerian integralsassociated with Kampe de Ferietrsquos functionrdquo Publications DeLrsquoInstitut Mathematique vol 9 no 23 pp 199ndash202 1969
[11] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Halsted Press Chichester UK
[12] E M Wright ldquoThe asymptotic expansion of the generalizedBessel functionrdquo Proceedings of London Mathematical Societyvol 38 pp 257ndash270 1935
[13] A Erdelyi Higher Transcendental Functions vol 1 McGraw-Hill New York NY USA 1953
[14] H M Srivastava K C Gupta and S P Goyal The HmdashFunctions of One and Two Variables With Applications SouthAsian Publishers New Delhi India 1982
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Engineering Mathematics 11
References
[1] A K Rathie ldquoA new generalization of generalized hypergeo-metric functionsrdquo Le Matematiche vol 52 no 2 pp 297ndash3101997
[2] I S Ansari F Yilmaz M S Alouni and O Kucur ldquoNewresults on the sum of Gamma random variates with applicationto the performance of wireless communication systems overnakagami-m fading channelsrdquo httparxiv-web3librarycornelleduabs12022576v4
[3] I S Ansari andYilmaz ldquoOn the sumof squared n-Randomvari-ates with application to the performance of wireless communi-cation systemsrdquo httparxiv-web3librarycornelleduabs12100100v1
[4] X Minghua W Yik-Chung and A Sonia ldquoExact outage prob-ability of dual-hop CSI-assisted AF relaying over nakagami-mfading channelsrdquo IEEE Transactions on Signal Processing vol60 no 10 pp 5578ndash5583 2012
[5] K ShanthaKumari T M Vasudevan Nambisan and AK Rathie ldquoA study of the I-function of two variablesrdquohttparxivorgabs12126717
[6] P K Mittal and K C Gupta ldquoAn integral involving generalizedfunction of two variablesrdquo Proceedings of the Indian Academy ofSciences A vol 75 no 3 pp 117ndash123 1972
[7] HM Srivastava and R Panda ldquoSome bilateral generating func-tions for a class of generalkized hypergeometric polynomialsrdquoJournal Fur die Reine und Angewandte Mathematik vol 17 no288 pp 265ndash274 1976
[8] B L J Braaksma ldquoAsymptotic expansions and analytic continu-ations for a class of Barnesintegralsrdquo Compositio Mathematicahvol 15 pp 239ndash341 1964
[9] Y L LukeThe Special Functions andTheir Approximations vol1 Academic Press New York NY USA 1969
[10] H M Srivastava and M C Daoust ldquoOn Eulerian integralsassociated with Kampe de Ferietrsquos functionrdquo Publications DeLrsquoInstitut Mathematique vol 9 no 23 pp 199ndash202 1969
[11] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Halsted Press Chichester UK
[12] E M Wright ldquoThe asymptotic expansion of the generalizedBessel functionrdquo Proceedings of London Mathematical Societyvol 38 pp 257ndash270 1935
[13] A Erdelyi Higher Transcendental Functions vol 1 McGraw-Hill New York NY USA 1953
[14] H M Srivastava K C Gupta and S P Goyal The HmdashFunctions of One and Two Variables With Applications SouthAsian Publishers New Delhi India 1982
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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