International Journal of Scientific and Research Publications, Volume 2, Issue 10, October 2012 1 ISSN 2250-3153 www.ijsrp.org Certain Case of Reducible Hypergeometric Functions of Hyperbolic Function as Argument Pooja Singh * , Prof. (Dr.) Harish Singh ** * Research Scholar, Department of Mathematics, NIMS University, Shobha Nagar, Jaipur (Rajasthan) ** Department of Business Administration, Maharaja Surajmal Institute, Affiliated to Guru Govind Singh Indraprastha University, New Delhi Abstract- In this paper, we specialized parameters and argument, Hypergeometric function F E (1 , 1 , 1 , 1 , 2 , 2 ; γ 1 , γ 2 , γ 3 ; cosh x, cosh y, cosh z) F G , F k and F N can be reduced to the hypergeometric function of Bailey’s F 4 (1 , 2 , γ 2 , γ 3 ; -coshy, - cosh z) and also discussed their reducible cases into Horn’s function. In the journal we consider hypergeometric func tion of three variables and obtain its interesting reducible case into Bailey’s F 4 & Horn’s function. In the section 2, hypergeometric function of four variables can be reduced to the hypergeometric function of one, two & three variables with some new and interesting particular cases. Index Terms- Matrix argument, Confluent, Hypergeometric function, Beta and Gamma integrals, M-transform I. ON HYPERGEMOTERIC INTEGRALS 1.1 INTRODUCTION e will study Laplace’s double integral for Saran’s function F E (1 , 1 , 1 , 1 , 2 , 2 ; γ 1 , γ 2 , γ 3 ; cosh x, cosh y, coshz) which has been reduced to Bailey’s F 4 (1 , 2 , γ 2 , γ 3 ; - cosh y, - cosh z) and pochhamer type of Integrals for F E , F G , F K and F N and also discussed their reducible cases into Horn’s functions. The purpose of studing only the function F E , F G , F K and F N is mainly due to the function in their integral representation contain Appell’s function F 1 or F 2 or the product of Gauss’s hypergeometric series which can be reduced by the following relations. 2 F 1 (, ; γ; cosh y) = m ! (γ (β n n n k,m ) ) ) ( 0 cosh m y (1.1.1) 2 F 1 (, ; γ; cosh y) = m m ! (γ e e (β n m y y n n k,m 2 ) ) ( ) ) ( 0 (1.1.2) 2 F 1 (, ; γ; cosh y) = 1 0 1 1 ) cosh 1 ( ) 1 ( ) ( ) ) ( dt y t t t (γ (1.1.3) 2 F 1 (, ; γ; cosh y) = (1- cosh y) -2 F 1 (,γ-; γ; 1 cosh cosh y y ) (1.1.4) Similarly 2 F 1 (, ; γ; cosh y) = (1- cosh y) -2 F 1 (γ - , ; γ; 1 cosh cosh y y ) (1.1.5) 2 F 1 (, ; γ; cosh y) = (1- cosh y) γ--2 F 1 (γ-, γ-; γ; coshy y) (1.1.6) 2 F 1 (, ; γ; cosh y) = (1- cosh y) -(1.1.7) F 1 (,,’,: cosh x, cosh y) = (1 – cosh y) -(1 – cosh y) -(1.1.8) W
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International Journal of Scientific and Research Publications, Volume 2, Issue 10, October 2012 1 ISSN 2250-3153
www.ijsrp.org
Certain Case of Reducible Hypergeometric Functions of
Hyperbolic Function as Argument
Pooja Singh*, Prof. (Dr.) Harish Singh
**
* Research Scholar, Department of Mathematics, NIMS University, Shobha Nagar, Jaipur (Rajasthan)
** Department of Business Administration, Maharaja Surajmal Institute, Affiliated to Guru Govind Singh Indraprastha University, New Delhi
Abstract- In this paper, we specialized parameters and argument, Hypergeometric function FE (1, 1, 1, 1, 2, 2; γ1, γ2, γ3; cosh x,
cosh y, cosh z) FG, Fk and FN can be reduced to the hypergeometric function of Bailey’s F4(1, 2, γ2, γ3; -coshy, - cosh z) and also
discussed their reducible cases into Horn’s function. In the journal we consider hypergeometric function of three variables and obtain
its interesting reducible case into Bailey’s F4 & Horn’s function.
In the section 2, hypergeometric function of four variables can be reduced to the hypergeometric function of one, two & three
variables with some new and interesting particular cases.
Index Terms- Matrix argument, Confluent, Hypergeometric function, Beta and Gamma integrals, M-transform
I. ON HYPERGEMOTERIC INTEGRALS
1.1 INTRODUCTION
e will study Laplace’s double integral for Saran’s function FE (1, 1, 1, 1, 2, 2; γ1, γ2, γ3; cosh x, cosh y, coshz) which has
been reduced to Bailey’s F4 (1, 2, γ2, γ3; - cosh y, - cosh z) and pochhamer type of Integrals for FE, FG, FK and FN and also
discussed their reducible cases into Horn’s functions. The purpose of studing only the function FE, FG, FK and FN is mainly due to the
function in their integral representation contain Appell’s function F1 or F2 or the product of Gauss’s hypergeometric series which can
be reduced by the following relations.
2F1 (, ; γ; cosh y) = m !(γ
(β
n
nn
k,m )
))(
0
cosh m y (1.1.1)
2F1 (, ; γ; cosh y) = mm !(γ
ee(β
n
myy
nn
k,m 2 )
)())(
0
(1.1.2)
2F1 (, ; γ; cosh y) =
1
0
11 )cosh1()1()()
)(dtyttt
(γ
(1.1.3)
2F1 (, ; γ; cosh y) = (1- cosh y)-
2F1 (,γ-; γ; 1cosh
cosh
y
y
) (1.1.4)
Similarly
2F1 (, ; γ; cosh y) = (1- cosh y)-
2F1 (γ - , ; γ; 1cosh
cosh
y
y
) (1.1.5)
2F1 (, ; γ; cosh y) = (1- cosh y)γ--
2F1 (γ-, γ-; γ; coshy y) (1.1.6)
2F1 (, ; γ; cosh y) = (1- cosh y)-
(1.1.7)
F1(,,’,: cosh x, cosh y) = (1 – cosh y) -
(1 – cosh y) -
(1.1.8)
W
International Journal of Scientific and Research Publications, Volume 2, Issue 10, October 2012 2 ISSN 2250-3153
K15 (a, a, a, b5; b1, b2, b3, b4; c, c, c, c; x, y, z, t)
=!!!!)(
)()()()()()( 43215
0,,, qpnm
tzyx
c
bbbbba qpnm
qpnm
qpnmqpnm
qpnm
(2.3.12)
=!!
)(
)(
)()()(
!!)(
)()()( 34521
0,,, qp
tzb
nmc
bbnma
nmc
yxbbaqp
p
qp
qqp
nm
nm
nmnm
qpnm
=
!!
)(
)(
)()()(
!!)(
)()()( 345
0,
21
0, qp
tzb
nmc
bbnma
nmc
yxbbaqp
q
qp
qqp
qpnm
nm
nmnm
nm
= F1( a, b1, b2; c ; x, y) F3(a+m+n, b5,b3,b4; c+m+n; z, t) (2.3.13)
Where F1 & F3 are Appell function of two variable
This completes the derivation of (2.3.4)
2.4 In this Section Quadruple hypergeometric function reduced to the Appell hypergeometric function Lauricella's set
and Saran
(1) Theorem By specializing the parameters of K2, K12, K15 , K6 we obtain the following
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K2 ( a, a, a, a; b, b, b, c; d1 d2, d3, d4; x,y,z,t)
= FE ( a, a,a, b,b,c,; d1, d2,d3 ; x,y,t) 2F1( a+m+n+q,b+m+n, d4; z ) (2.4.1)
K2, (a, a, a, a; b, b, b, c; d1, d2, d3, d4; x,y,l,t)
)()(
)22()(
44
44
nmbdqnmad
nmqbadd
FE (a,a,a,b,b,c,; d1,d2,d3; x, y, t) (2.4.2)
K12 ( a, a, a, a; b1,b2, b3,b4, c1, c1, c2, c2; x,y,z,t)
= FG( a, a, a, b1,b3, b4, c1,c2, c2; x,z,t) 2F1 (a+m+p+q,b2;c1+m;y) (2.4.3)
K12 ( a, a, a, a; b1, b2, b3, b4, c1, c1,c2, c2 ; x,l,z,t)
)()(
)()(
121
211
qpacbmc
qpbacmc
FG ( a, a, a, b1,b3, b4,; c1, c2, c2; x, y, z, t ) (2.4.4)
K15 ( a, a, a, b5;b4, b1,b2,b3; c, c, c,c; x, y, z, t )
= FS( b5, a, a, b4,b1,b2;c,c,c; x,y,t) 2F1 (a+m+n,b3 ;c+q+m+n; z) (2.4.5)
K15 ( a, a, a, b5; b4, b1, b2, b3;c, c, c, c; x, y, z, t)
)()(
)()(
3
3
aqcbnmqc
baqcnmqc
FS(b5, a, a, b4, b1, b2,; c,c,c; x,y,t) (2.4.6)
K6 (a,a,a,a; b,b, c1,c2;e, d,d,d;x,y,z,t)
= FF( a, a, a,b, c1, b, ;e,d,d; x,z,y) 2F1 ( a+m+p+n,c2 ;d+p+n; t) (2.4.7)
K6 ( a,a,a,a; b,b, c1,c2;e, d,d,d;x,y,z,t)
=)()(
)()(
2
2
cnpdmad
camdnpd
FF( a, a, a,b, c1, b, ;e,d,d; x,z,y) (2.4.8)
Where (F4,F14, F8,F7) & (FE, FF, FG,FS) are Lauriceila's set & Saran Triple hypergeometric Series.
(1) Proof:- Now Quadruple hypergeometric function can be reduced to Lauricella's set and Saran Triple hypergometric Series.
K2 ( a,a,a,a; b,b,b,c; d1d2,d3,d4;x,y,z,t)
=!!!!)()()()(
)()()(
43210,,, qpnm
tzyx
dddd
cba qpnm
qpnm
qpnmqpnm
qpnm
(2.4. 9)
=!
)(
)(
)(
!!)()()(
)()()(
34210,,, p
znmb
d
qnma
nmddd
tyxcba p
p
p
p
qnm
qnm
qnmqnm
qpnm
=
!)(
))((
!!!)()()(
)()()(
304210,, p
z
d
nmbqnma
qnmddd
tyxcba p
ppqnm
qnm
qnmqnm
nmq
= FE ( a, a, a, b, b,c,; d1 , d2, d3 , x,y, t) 2F1 ( a+m+n+q, b+m+n, d4; z) (2.4.10)
This completes the derivation of (2.4.1)
(2) Proof:
If z = 1, in euqation (2.4.10)
K2 (a, a, a, a; b, b, b, c; d1, d2, d3, d4; x,y, l,t)
= FE( a, a, a, b, b,c,; d1, d2, d3, x,y, t) 2F1 ( a+m+n+q, b+m+n, d4; 1) (2.4.11)
Now Apply Gauss's summation theorem in equation ( 2.4.11)
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F1(,,γ;1)= )()(
)()(
K2 ( a, a, a, a; b, b, b, c; d1, d2, d3, d4; x,y, l,t)
=)()(
)22()(
44
44
nmbdqnmad
nmqbadd
FE( a, a, a,b,b, c,; d1,d2,d3; x,z,t) (2.4.12)
This completes the derivation of (2.4.2)
(3) Proof: K12 ( a, a, a, a; b1, b2, b3,b4,c1,c2, c2 ; x,y, z,t)
=!!!!)()(
)()()()()(
21
4321
0,,, qpnm
tzyx
cc
bbbba qpnm
qpnm
qpnmqpnm
qpnm
(2.4.13)
=!)(!!!
)()(
)()(
)()()()(
1
2
21
431
0,,, nmcqpm
ybqpma
cc
tyxbbba
n
n
nn
qpm
qnm
ppmqpnm
qpnm
=
!)(
)()(
!!!)()(
)()()()(
1
2
021
431
0,, nmc
ybqnma
qpmcc
tyxbbba
n
n
nn
pqpm
qpm
qpmqnm
qpm
FG ( a, a, a, b1, b3, b4; c1, c2,c2; x, z, t) 2F1 (a+m+p+q; b2; c1+m; y) (2.4.14)
This completes the derivation of (2.4.3)
(4) Proof:
When y = 1, in equation (2.4.14) Then
K12 ( a, a, a, a; b1, b2, b3, b4, c1, c1, c2, c2; x,I, z,t)
=FG (a, a, a, b1, b3, b4; c1, c2,c2; x, z, t). 2F1 (a+m+p+q; b2; c1+m; 1) (2.4.15)
Now Apply Gauss's summation theorem in equation ( 2.4.15)
F1(,,γ;1)= )()(
)()(
=)()(
)()(
121
211
qpacbmc
qpbacmc
FG( a, a, a,b1,b3, b4,c1, c2, c2,; x,z,t) (2.4.16)
This completes the derivation of (2.4.4)
(5) Proof :- K15 (a, a, a, b5; b1, b2, b3, b4 ; c, c, c, c; x; y, z, t)
=!!!!)(
)()()()()()( 43215
0,,, qpnm
tzyx
c
bbbbba qpnm
qpnm
qpnmqpnm
qpnm
=!)(!!!)(
)()()()()()()( 32145
0,,, pnmqcnmqc
zbnmayxtbbbab
pnmq
q
pp
nmq
nmqnmq
qpnm
=
!)(
)()(
!!!)(
)()()()()(2
0
2145
0,, nnmqc
ybqppma
pmqc
yxtbbbab
n
n
nn
pnm
nmq
nmqnmq
qpm
= FS( b5, a, a, b4,b1,b2;c,c,c; x,y,t) 2F1 ( a+m+n,b3 ;c+q+m+n; z ) (2.4.17)
This completes the derivation of (2.4.5)
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(6) Proof :- When z = 1 in equation (2.4.17) Then
K15 (a, a, a, b5, b4, b1, b2, b3; c, c, c, c; x, y, 1, t )
= Fs( b5, a, a, b4,b1,b2;c,c,c; x,y,t) 2F1 ( a+m+n,b3 ;c+q+m+n;1) (2.4.18)
Now Apply Gauss's summation theorem in equation
F1(,,γ;1)=)()(
)()(
=)()(
)()(
3
3
aqcbnmqc
baqcnmqc
FS(b5,a, a,b4,b1, b2,c, c, c,; x,z,t) (2.4.19)
This completes the derivation of (2.4.6)
(7) Proof:- K6 ( a, a, a, a; b, b, c1,c2 ; e, d, d, d; x, y, z, t)
=!!!!)()(
)()()()( 21
0,,, qpnm
tzyx
de
ccba qpnm
qpnm
qpnmqpnm
qpnm
(2.4.20)
=!!!!
)()(
)()(
)()()( 21
0,,, qpnm
tcnpma
de
yzxcba q
qq
qpnm
npm
pnmnpm
qpnm
=
!)(
)()(
!!!)()(
)()()( 2
0
1
0,, qnpd
tcnpma
npmde
yzxcba
q
q
qq
pnpm
npm
pnmnpm
qpm
= FF( a, a, a,b, c1, b, ;e,d,d; x,z,y) 2F1 ( a+m+p+n,c2 ;d+p+n; t) (2.4.21)
This complete the derivation of (2.4.7)
(8) Proof :-
When t = 1 in equation (2.4.21) Then
K6 ( a, a, a, a; b, b, c1,c2; e, d, d, d; x, y, z, 1)
= FF( a, a, a,b, c1, b,; e,d,d; x,z,y) 2F1 ( a+m+p+n,c2 ;d+p+n; 1) (2.4.22)
Now Apply Gauss's summation theorem in equation
F1(,,γ;1)= )()(
)()(
=)()(
)()(
2
2
cnpdmad
camdnpd
FF(a, a,a,b,c1, b,;e,d,d; x,z,y) (2.4.23)
This completes the derivation of (2.4.8)
REFERENCES
[1] APPELL, P. (1880): Surles series hypergeometric de deux variable, et surds equationa diiferentiells linearies aux derives partielles, C.R. Acad. Sci. Paris, 90, 296-298.
[2] ERDELYI, A. (1948) : Transformation of the hypergeometric functions of four variables, Bull soco grece (N.S.) 13, 104-113.
[3] EXTON, H. (1972) : Certain hypergeometric functions of four variables, Bull soco grece (N.S.) 13, 104-113.
[4] HORN, J. (1931) : Hypergeometric Funktionen Zweier Veranderlichen Math. Ann, 105, 381 – 407.
International Journal of Scientific and Research Publications, Volume 2, Issue 10, October 2012 18
ISSN 2250-3153
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[5] SARAN, S. (1955) : integrals associated with hypergoemetric functions of there variables, Not. Inst. of Sc. of India, Vol. 21, A. No. 2, 83-90.
[6] SARAN, S.(1957): Integral representations of laplace type for certain hypergeometric functions of three variables, Riv. Di. Mathematica, parma 133-143.
[7] WHITTAKER, E.T. AND WATSON, G.N. (1902) : A course of Modern Analysis
AUTHORS
First Author – Pooja Singh, Research Scholar, Department of Mathematics, NIMS University, Shobha Nagar, Jaipur (Rajasthan)