The Meixner Polynomials in Several Variables · ber theory, combinatorics, stochastic processes. On . the other hand, generating functions have a great importance in special functions

The Meixner Polynomials in Several Variables

Nejla Özmen1*, Esra Erkus-Duman2 1 Düzce University, Faculty of Art and Science, Department of Mathematics, Konuralp TR-81620, Düzce, Turkey. 2 Gazi University, Faculty of Science, Department of Mathematics, Teknikokullar TR-06500, Ankara, Turkey. * Corresponding author. Tel.: +903805421100; email: [email protected], [email protected] Manuscript submitted February 12, 2019; accepted April 9, 2019.

Abstract: The polynomials are applied in many areas of mathematics, for instance, continued fractions,

operator theory, analytic functions, interpolation, approximation theory, numerical analysis, electrostatics,

statistical quantum mechanics, special functions, number theory, combinatorics, stochastic processes. On

the other hand, generating functions have a great importance in special functions theory. The present study

deals with some new properties of the Meixner polynomials in several variables. Firstly, we obtained some

results include various families of multilinear and multilateral generating functions for the Meixner

polynomials in several variables. In the last section, we get a theorem that gives a bilateral generating

functions for these polynomials and the Lauricella functions. Finally, we derive some corollaries of the last

theorem.

Key words: Bilinear and bilateral generating function, Lauricella function, Meixner polynomials.

1. Introduction

The Meixner polynomials are denoted by ),;( cxmn and are defined as [1]

xxn

n

n c

tt

n

tcxm )1()1(

!),;(

0

(1)

where ,0 10 c and ,...2,1,0x .

It is from (1) that (see [1]):

,11

)()!(!

!)()(),;(

0

k

kk cknk

nxcxm kn

n

n

where v denotes the Pochhammer symbol.

Definition 1: The Meixner polynomials of s-variables are defined by [2]:

,

)!()(

)1()()()(

...

),...,,;,...,(

1

1

11...

11

...1

...1

00201

111

j

s

j

j

s

jj

s

jnrrnrnn

r

ssn

r

cxn

ccxxm

srr

jr

jrsrr

srr

s

(2)

International Journal of Applied Physics and Mathematics

144 Volume 9, Number 3, July 2019

doi: 10.17706/ijapm.2019.9.3.144-151

mailto:[email protected]

mailto:[email protected]

where ,0 ,10 jc , . . .2,1jx and .,...,2,1 sj

Theorem 1: (see [2]) The following generating functions holds true for the Meixner polynomials of

s-variables defined by (2):

.,...,,1min

)1()1(!

),...,,;,...,(

1

1

...

11

0

1

s

x

j

s

j

xxn

ssn

n

cct

c

tt

n

tccxxm js

(3)

Theorem 2: (see [2]) The following generating function relationship for the Meixner polynomials of s

variables holds true:

1

1 1

0

... 11

1

( ,..., ; , ,..., )!

(1 ) 1 ,..., ; , ,..., .1 1

j

s

n

n k s s

n

xs

k x x sk s

j j

tm x x c c

n

c tc ttt m x x

c t t

(4)

In Section 2, we establish several theorems involving various families of generating functions for the

Meixner polynomials ),...,,;,...,( 11 ssn ccxxm by applying the method which was discussed by Chen and

Srivastava [3]. In the last section, we derive several families of bilateral generating functions for the

multivariable Meixner polynomials and the generalized Lauricella functions.

2. Bilinear and Bilateral Generating Functions

In this section, we derive several bilinear and bilateral generating functions for the Meixner polynomials

),...,,;,...,( 11 ssn ccxxm which generated by (2) and given explicitly by (3) using the similar method

considered in [4]-[7].

Theorem 3: Let

.)!(

),...,(),...,,;,...,(:

;,...,;,...,,;,...,

111

/

0

111

,

,

pknyyccxxma

yyccxx

k

rksspknk

pn

k

rsspn

(5)

If

k

rkk

k

r yyayy ),...,(:);,...,( 1

0

1,

then, for every nonnegative integer , we have

n

prsspn

n

tt

yyccxx

;,...,;,...,,;,..., 111

,

,

0

(6)



).;,...,()1()1( 1,

1

...21

r

x

j

s

j

xxxyy

c

tt js

Proof: If we denote the left-hand side of (6) by T and use (5),

.)!(

),...,(),...,,;,...,( 111

]/[

00 pkn

tyyccxxmaT

pknk

rksspknk

pn

kn

Replacing n by ,pkn

!),...,(),...,,;,...,( 111

00 n

tyyccxxmaT

nk

rkssnk

kn

k

rkk

k

n

ssn

n

yyan

tccxxm ),...,(

!),...,,;,...,( 1

0

11

0

);,...,()1()1( 1,

1

...21

r

x

j

s

j

xxxyy

c

tt js

which completes the proof.

Theorem 4: For a non-vanishing function ),...,( 1 ryy of complex variables ryy ,...,1 )( Nr , let

.,0 ),...,(),...,,;,...,(

);,...,;,...,,;,...,(

111

0

111,,

C

n

n

rpnssqnmn

n

rssqp

atyyccxxma

tyyccxx

Suppose also that

., )!(

),...,();,...,( 1

/

0

1,, C

pqkn

zyyazyy

k

rpkk

qn

k

rqpn

Then, for Nqp, , we have

.)1

(;,...,;1

,...,1

,;,...,)1()1(

);,...,(),...,,;,...,(

11

1,,

1

...

1,,11

0

21

q

rs

sqp

x

j

s

j

xxxm

n

rqpnssnm

n

t

tzyy

t

tc

t

tcxx

c

tt

tzyyccxxm

js

(7)

Proof: Let S denote the first member of the assertion (7) of Theorem 4. Then,

.)!(

),...,(),...,,;,...,( 1

/

0

11

0 qkn

tzyyaccxxmS

nk

rpkk

qn

k

ssnm

n

Now, setting n by qkn and then using relation (4),



!),...,(),...,,;,...,( 111

00 n

tzyyaccxxmS

qknk

rpkkssqknm

kn

kq

rpkk

n

ssqknm

nk

ztyyan

tccxxm ))(,...,(

!),...,,;,...,( 111

00

jsx

j

s

j

xxqkm

k c

tt )1()1(

1

...

0

1

kq

rpkks

sqkm ztyyat

tc

t

tcxxm ))(,...,(

1,...,

1,;,..., 1

11

jsx

j

s

j

xxm

c

tt )1()1(

1

...1

q

rs

sqpt

tzyy

t

tc

t

tcxx )

1(;,...,;

1,...,

1,;,..., 1

11,,

which completes the proof.

3. The Generalized Lauricella Functions

In the present section, we derive various families of bilateral generating functions for the Meixner

polynomials in several variables and the Lauricella functions. Some of the definitions and notations used in

this paper are presented here as follows. A further generalization of the familiar Kampé de Fériet

hypergeometric function in two variables is due to Srivastava and Daoust [8] who defined the

Srivastava-Daoust (or generalized Lauricella) function as follows:

,!

...!

),...,(

;:)(...;

,...,

;:)(...;

;:)(:,...,:)(

;:)(:,...,:)(

1

11

0,...,

)()(

1

)()(

)1()1()()1(

)1()1()()1(

1

1

)(;...;)1(:

)(;...;)1(:

n

m

n

m

n

mm

nn

n

nn

n

n

m

z

m

zmm

d

zz

b

dc

ba

F

n

n

nBBA

nDDC

where for convenience

),...,( 1 nmm

)(

)(

)(

)(

)1(1

)1(

)1(1

)1(

)()1(1

)()1(1

)(

)(

...

)(

)(

)(

)(

:)(

1

)(

1

)1(

1

)1(

1

...1

...1

njn

n

njn

n

j

j

njnj

njnj

m

n

j

D

j

m

n

j

B

j

mj

D

j

mj

B

j

mmj

C

j

mmj

A

j

d

b

d

b

c

a

the coefficients

),,...,1 ;,...,1( and ),,...,1 ;,...,1( )()()( nkBjnkAj kk

j

k

j

),...,1 ;,...,1( and ),,...,1 ;,...,1( )()()( nkDjnkCj kk

j

k

j



are real constants and )(

)(

k

kBb abbreviates the array of

)(kB parameters [9],

).,...,1 ;,...,1( )()( nkBjb kk

j

For a suitably bounded non-vanishing multiple sequence 0N,...,1

),...,( 1

nmmnmm of real or

complex parameters, we define a function ),...,;( 21 sn uuu of s variables suuu ,...,; 21 (real or

complex) be defined by [9],

!...

!,...,),,...,(

))((

))(()(:

),...,;(

1

121

21

1

1

11

0,...,201 s

sss

n

sn

m

u

m

ummmmf

d

bn

uuu

smm

m

mm

smmm

(8)

where, for convenience

.)())(( , )())((1111

11jmmjmm j

D

j

j

B

j

ddbb

Theorem 5: The following bilateral generating function holds true:

,

!...

!!...

!!

))((

)...()()...())((

,...,),,...,...(

1)1(

!),...,;(),...,,;,...,(

211

(

11(

211

1

...

2111

0

22

)1

(11

1)

1

11(1

11

11

)...11

1)...11

0,...,2,,...,1,1

1

1

rs

sks

rrs

x

j

s

j

xx

n

rnssn

n

mmkkm

d

xxkkb

mmmkkmf

c

tt

n

tuuuccxxm

rmru

mu

sk

tscsc

t

tuk

tc

c

t

tum

t

tu

skkm

skmskkm

rmmskkm

j

s

where ),...,;( 21 rn uuu is given by (4).

Proof: By using the relationship (4), we can easily derive the following generating functions

!),...,;(),...,,;,...,( 2111

0 n

tuuuccxxm

n

rnssn

n

1

11

21))((

))(()(),...,,;,...,(

0,...,0

11

0 m

mm

mm

n

m

ssn

n d

bnccxxm

r



!!

...!

,...,),,...,(1

121

1

n

t

m

u

m

ummmmf

n

r

m

r

m

rr

r

j

s

x

j

s

j

xx

c

tt

1)1(

1

...1

)...(

11)...(

0,...,,,...,,11

1111

211))((

)...()()...())((

s

ss

rs kkm

kskmskkm

mmkkm d

xxkkb

rrs mmmmkkmf ,...,),,...,,...( 2211

.

!...

!!

)(...

!

)(

! 2

2

1

1

1

1

1

1

121

1

1

1111

r

m

r

m

s

k

tc

c

t

tuk

tc

c

t

tum

t

tu

m

u

m

u

kkm

rs

s

s

Corollary 1: Upon setting

r

rr

mrm

mrmmm

rrcc

bbammmmf

)...()(

)...()()(,...,),,...,(

1

21

1

2...

21

and 0 in Theorem 5,

!

,...,;,...,;,...,,,),...,,;,...,( 112

)(

11

0 n

tuuccbbnaFccxxm

n

rrr

r

Assn

n

1;...;1;1;...;1;1:1

1;...;1;0;...;0;0:1

1

...1)1( 1 F

c

tt

j

s

x

j

s

j

xx

;...;;;:,...,:)(

];1:[];...;1:[];1:...[:1,...,1:)(

)()1(

1

11

rs

ss

c

xxkka

ruutscsc

t

tu

tc

c

t

tu

t

tu

rr

rr

dd

bb

,...,2,)1

(11

,...,)1

11(1

1,1

1

];:)[(...;];:)[(

];:)[(...;];:)[(

)()()2()2(

)()()2()2(

where )(s

AF the Lauricella function and the coefficients )( are given by

.

10

)11(1)(

rss

s

Corollary 2: If we put

rmm

rmmrmm

c

aaaammmmf

rr

rr

...

1

2

)(

)...()()...()(,...,),,...,(

)1(

2

)1(

2

)1(

1

)1(

1

21

2

and bbB 11 and 0 ,1 ,1 in Theorem 5,



,

;. . . ;;

;1:. . . ;;1:

;. . . ;;:1, . . . ,1:)(

];1:[];...;1:[];1:...[:,...,:

1)1(

!,...,;;,...,,,,...,,),...,,;,...,(

,...,2,)1

(11

,...,)1

11(1

1,1

1

)1()1(

11

)()1(

2;...;2;1;...;1;1:1

0;...;0;0;...;0;0:1

1

...

1

)1(

2

)1(

2

)1(

1

)1(

1

)(

11

0

1

ruutscsc

t

tu

tc

c

t

tu

t

tu

r

ss

rs

x

j

s

j

xx

n

r

rrr

Bssn

n

aa

c

xxkkb

Fc

tt

n

tuucaabaanFccxxm

j

s

where )(s

BF the Lauricella function and the coefficients )( are given by

.

1,0

11,1)(

rss

s

Corollary 3: Finallay, if we let

r

rr

mm

mrmmm

rrc

bbammmmf

...

2...

21

1

21

)(

)...()()(,...,),,...,(

and 0 in Theorem 5, we have the following result:

!

,...,,;;,...,,,),...,,;,...,( 212

)(

11

0 n

tuuucbbnaFccxxm

n

rr

r

Dssn

n

;;,...,,,...,,...,1)1( 211

)(

1

...1 cbbxxkkaFc

tt rss

rs

D

x

j

s

j

xx

j

s

.,...,,)1

(1

,...,)1

(1

,1

21

1

111

r

s

s uutc

c

t

tu

tc

c

t

tu

t

tu

References

[1] Chihara, T. S. (1978). An Introduction to Orthogonal Polynomials. New York: Gordon and Breach Science

Publishers Inc.

[2] Khan, M. A., & Akhlaq, M. (2012). A note on generating functions and summation formulae for Meixner

polynomials of several variables. Demonstratio Mathematica, 1, 51-66.

[3] Chen, K.-Y., & Srivastava, H. M. (2005). Series identities and associated families of generating functions.

J. Math. Anal. Appl., 311, 582-599.

[4] Erkus-Duman, E., Altın, A., & Aktas, R. (2011). Miscellaneous properties of some multivariable

polynomials. Math. Comput. Modelling, 54, 1875-1885.

[5] Özmen, N., & Erkus-Duman, E. (2018). Some families of generating functions for the generalized Cesáro



polynomials. J. Comput. Anal. Appl., 25, 670-683.

[6] Korkmaz-Duzgun, D., & Erkus-Duman, E. (2018). The laguerre type d-orthogonal polynomials. J. Sci.

Arts, 42, 95-106.

[7] Özmen, N. (2017). Some new properties of the Meixner polynomials. Sakarya University Journal of

Science, 21(6), 1454-1462.

[8] Srivastava, H. M., & Daoust, M. C. (1969). Certain generalized Neumann expansions associated with the

Kampé de Fériet function. Nederl. akad. Westensch. Indag. Math, 31, 449-457.

[9] Liu, S.-J., Lin, S.-D., Srivastava, H. M., & Wong, M.-M. (2012). Bilateral generating functions for the

Erkuş-Srivastava polynomials and the generalized Lauricella functions. App. Mathematcis and Comp.,

218, 7685-7693.

Nejla Özmen is an assistant professor of mathematics at Düzce University in Düzce,

Turkey. In 2012 she received the Ph.D. degree from the Department of Mathematics in

Gazi University. She has been working in Düzce University since 2009. Her research areas

contain special functions, generating functions.

Esra Erkus-Duman is a full professor of mathematics at Gazi University in Ankara, Turkey.

In 2005 she received the Ph.D. degree from the Department of Mathematics in Ankara

University. Before she came to Gazi University, she worked as a research assistant at

Ankara University. She became associate professor in 2010 and full professor in 2017.

Her research areas contain special functions, generating functions and orthogonal

polynomials.



The Meixner Polynomials in Several Variables · ber theory, combinatorics, stochastic processes. On . the other hand, generating functions have a great importance in special functions

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