Certain Bilateral Generating Relations for a Class of Generalized Hypergeometric Functions of Two Variables

Certain Bilateral Generating Relations for a Class of

Generalized Hypergeometric Functions of Two Variables

B.Satyanarayana1,*_{, N. Srimannarayana}2 _{,Y. Pragathi Kumar}1

1_{Department of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar-522 510, Andhra Pradesh, India} 2_{Department of Science and Humanities, RISE Gandhi Group of Institutions, Ongole -523 272, Andhra Pradesh, India}

*Corresponding Author: [email protected]

Copyright © 2014 Horizon Research Publishing All rights reserved.

Abstract

In [10] we defined and studied a class of generalized hypergeometric functions ( , )

B

_nα β

( , , )

x y w

. In this paper an attempt has been made to obtain some bilateral generating relations with ( , )

B

_nα β

( , , )

x y w

. Each result is followed by its applications to the classical orthogonal polynomials.

Keywords

Bilateral Generating Relations, Generalized Hypergeometric Functions, Classical Orthogonal Polynomials

1. Introduction

In the previous paper [10], we introduced a class of generalized hypergeometric functions ( , )

B

_nα β

( , , )

x y w

defined as follows:

(

1) (

1)

( , )( , , )

₂

( !)

n

n

B

_n

x y w

n

α

β

α β

₌

Γ + + Γ + +

_×

[ ] 0

( 1)

_{( , )}

! (

1) (

1)

n r rw

n r r

y

_J

_{x w}

r

n

r

r

α

α

β

−

=

−

Γ + − + Γ

+ +

∑

(1.1) where

J x w

_nα

( , )

is modified Jacobi polynomial (see Parihar and Patel [6] and also see Lahiri and Satyanarayana [3]-[5]). We also derived the following relation

( , )

2

(1

) (1

)

( , , )

( !)

n n

n

B

x y w

n

α β

₌

+

α

+

β

_×

_,

1:1;1

:1;1

:

; ;

,

:1

;1

;

y x

n

F

w w

w w

β

α

−



_{− −}





₋







−− +

+





(1.2)

Taking the limit

w

→

0

in (1.1), we obtain

( , ) ( , ) 0

lim

_n

( , , )

_n

( , )

w

B

x y w

L

x y

α β α β

→

=

, (1.3)

where

L

( , )_nα β

( , )

x y

is Laguerre polynomial of two variables [7].

In Satyanarayana [9] ( also see [5, p.326(1.8)] defined generalized hypergeometric functions _{; ;( )}; ;( )p

( , )

q

a n b

I

α µ_λ

x w

and also proved that [9, p.65(3.3.3)]

; ;( ) ; ;( ) 0

(

)

_{( , )}

(1

)

p q

a _n

n

n r b n

n

n r

r

_I

_{x w t}

r

r

α µ λ

ρ

α

∞

+ =

+





+



_{ + +}





∑

(1 )

t

r

r

r

ρ

α

− −



+



= −

_

_





(3)

( ) ::

1; ; ;

( ) ::1

; ; ;

p

q

x

a

w

F

b

µ

α



_{− + −− −−}





+ −− −−



; ;

, ,

1

; ; ;

x

r r

_{wt w w}

w

_t

ρ

+ −

−

+

λ



_

−



−

−− −− −−

_

(1.4)

where

F

(3) is generalized hypergeometric functions of three variables (see Srivastava and Karlsson [11]).

In particular for

λ

=

0

and

µ

=

1,

we have

;1;( )

;0;( )pp

( , ) (1

)

( , )

x

a _{w n}

n a

I

α

x w

= −

w J x w

α , (1.5)

where

J x w

_nα

( , )

is modified Jacobi polynomial (see Parihar and Patel [6]) and

; ;( ) ; ;( ) 0

lim

p

( , )

( ),

p

a _x

n n a

w

I

x w e L x

α µ _α

λ −

→

=

(1.6)

where

L x

α_n

( )

is Laguerre polynomial [8].

p.302] Gottlieb polynomial 2 1

( ; )

n

( , ;1;1

)

n

x

e

λ

F n x

e

λ

φ

λ

=

−

− −

−

_{, (1.7)}

Generalized Sylvester polynomial

2 0

( )

1

( ; )

( , ; ;

)

!

n

n

ax

f x a

F

n x

n

ax

=

−

− −

(1.8) Agarwal and Manocha [1, p.1372 (2.2)(5.5)]

0

( ; )

n

(1 )

x k

n k n

n k

x

t

t

k

φ

λ

∞

− +

=

+





= −

×









∑

1

(1

−

te

− − −λ

)

x

( ;log

)

1

k

x

e

e

_t

t

λ

φ





_

−





_

−





(1.9) and

0

( ; )

n

(1 )

x k

n k n

n k

f

x a t

t

k

∞

− − +

=

+





= −

×









∑

( ; (1 ))

axt k

e

f x a

−

t

(1.10)

2. Main Results

Bilateral generating relations

We have derived the following bilateral generating relations for the class of generalized hypergeometric functions

( , )

_{( , , )}

n

B

α β

x y v

:

[

]

2 2 0

( ) ( !)

(1

)

(1

)

n

n n n

n

ρ

α

β

∞

=

×

+

+

∑

( , )_{( , , ) ; ;( )}; ;( )p

( , )

q

a _n

n _{x y v n b}

B

α β

I

α µ_λ

x w t

2:0;0;1;2;1 2:0;0;0;1;1

[( ) :1,1,1,0,1],

(1 )

[( ) :1,1,1,0,1],

p p

q

q

a

t

F

b

ρ +

− +





= −

_





[ :1,1,0,1,1],[

1:1,1,0,0,1]:

[1

:1,1,0,0,1],[ : 0,1,0,1,1,]:

x

w

ρ

µ

α

ρ

− +

+

; ;

; ,

;

; ; ;1

;

x

y

w

λ ρ

v

β

− − − +

−

− − − +

;

,

, ,

,

1

1

1

;

x

wt

_{w w}

vt

_wv

v

t

t

α



−

_

−



−

−

+

_

(2.1)

Where F is a generalized Lauricella hypergeometric function of 5 variables.

2

( , ) ( , , ) 0

( !)

(1

) (1

_n

)

_n n x y v

n

n

_B

α β

α

β

∞

=

×

+

+

∑

( ; )

n

n

x

t

φ

λ

1

(1 ) (1

t

x

te

− − −λ

)

x

= −

−

×

3:0;0;1

2:0;0;1

[1:1,1,1],[

:1,1,0],[ :1,0,1]:

[1

:1,1,1], [1:1,0,1] :

y

_x

F

v

β



₋

₋





+



1 2 3

; ; / ;

, ,

; ;1

;

x v

t t t

α

− −





− − +

_

(2.2)

Where F is a generalized Lauricella hypergeometric function of 3 variables.

1

(

1) ,

2

(

)(1 )

v e

t

vt

t

t

e

t

t

e

t

λ

λ λ

−

=

=

−

−

−

( )

( )

2 3

1

.

1 (1 )

v e

t

t

e

t

λ

λ

−

=

−

−

2

( , ) 0

( !)

_{( , , )}

(1

) (1

_n

)

_n n

n

n

_B

α β

_{x y v}

α

β

∞

=

×

+

+

∑

( ; )

n

n

f x a t

(1 )

_1:0;0;12:0;0;1

[

:1,1,1],

[1

:1,1,1],

x axt

y

t

e F

v

β

−



_

−

= −



+



2

[ :1,0,1]: ; ; ;

_,

_,

1

1

: ; ;1

;

n

x

x

_v

vt

_vtax

v t

t

t

α







− −





_

−

_

_



₋



_

₋

_





_

_

−

− − +

_

_(2.3)

Proof of (2.1). From (2.1), we have

[

]

2 2 0

( ) ( !)

(1

)

(1

)

n

n n n

n

ρ

α

β

∞

=

×

+

+

∑

( , )

( , , )

( , )

n n x y v n

0 0 0

( )

( )

(1

) (1

) (1

)

r s n r n

n r

n s r

n s r

y

n

v

ρ

α

α

β

+ ∞ − = = =





−

_

−

_





=

×

+

+

+

∑ ∑ ∑

( )

( , )

!

!

r s n s n

x

_v

v

_{v I x w t}

r

s

α

  −

 

 

0 0

( ) ( )

( )

(1

) (1

) (1

) !

r s s r

n

r s

r s r

s r

y

x

n

vt v

v

v

s

ρ

α

α

β

∞ = =



  

−

_

−

_{  }



  

=

×

+

+

+

∑ ∑

0

(

)

_{( , )}

(1

n

)

_n n r n

n

n r

r

_I

_{x w t}

r

r

α

ρ

α

∞ + =

+





+



_{ + +}





∑

By using (1.4), we get

0 0

( )

( ) ( )

(1

) !

(1

) (1

)

s r

s r

n

s r

s r r

s r

x

y

n

v

vt

v

v

s

ρ

α

α

β

∞ = =

 





−

_{ }

_

−

_

 





=

+

+

+

∑

∑

(3)

( ) ::

(1 )

( ) ::

p r q

a

r

t

F

r

b

ρ

α

− −



+





−

_

_







_

1; ; ;

; ;

;

, ,

1

1

; ; ;

; ;

;

x

_{r r}

x

wt w w

w

w

t

µ

ρ

λ

α



− + − − + − − + −

_



−

+

− − − −

−

_

, , , , 0

( )

(1 )

( )

p m n k s q m n k s m k n r s

a

t

b

ρ ∞ + + +

−

+ + + =

= −

∑

×

( )

1

(1

)

( )

m n r s

m n s m n s n r s

x

w

ρ

µ

α

ρ

+ + + + + + + + +



_{− +}











_×

+

( )

(1

) (1

)

r

k r s

s r

x

y

x

w

λ

v

ρ

v

α

β



_{− +}

 

₋



 



 



 



 



_{  ×}

+

+

( )

1

( )

1

! ! !

!

!

r

m n k

vt

s

wt

w w

t

wv

t

m n k

r

s









−

−

−











₋







2:0;0;1;2;1 2:0;0;0;1;1

[( ) :1,1,1,0,1],

(1 )

[( ) :1,1,1,0,1],

p p q q

a

t

F

b

ρ + − +



= −





[ :1,1,0,1,1],[

1:1,1,0,0,1] :

[1

:1,1,0,0,1],[ : 0,1,0,1,1,]:

x

w

ρ

µ

α

ρ

− +

+

; ;

; ,

; ;

; ; ;1

;1

;

x

y x

w

λ ρ

v v

β

α

− − − +

−

− − − +

+

,

, ,

,

1

1

wt

_{w w}

vt

_wv

t

t

−

₋





−

−



Hence complete the proof of (2.1). Applications.

(i) By setting p = q,

a = b ,

_{j j} j = 1, 2… p,

μ = 1

and

λ = 0

in (2.1), we get

[

]

2 2 0

( ) ( !)

(1

)

(1

)

n

n n n

n

ρ

α

β

∞ =

×

+

+

∑

( , )

( , , )

( , )

n

(1 )

n x y v n

B

α β

J x w t

α

= −

t

−ρ

×

2:0;0;2;1

2:0;0;1;1

[ :1,1,1,1],[ :1,1,0,1]:

[1

:1,1,0,1],[ : 0,1,1,1,]:

x

F

ρ

w

α

ρ







+



; ; ,

; ;

_,

_,

_,

1

1

; ;1

;1

;

y x

_wt

_vt

w

wv

v v

_t

_t

ρ

β

α



− −

−

−

₋

_



−

−

− − +

+

_

(2.4)

(ii) Applying p = q,

a = b ,

_{j j} j = 1, 2, …, p and writting

0

w

→

in (2.1), we get

[

]

2 ( , ) ( , , ) 2 0

( ) ( !)

_{( )}

(1

)

(1

)

n n

n x y v n

n n n

n

_B

α β

_{L x t}

α

ρ

α

β

∞ =

+

+

∑

1:0;0;2;1

2:0;0;1;1

[ :1,1,1,1]

:

(1 )

[1

:1,1,0,1],[ : 0,1,1,1,]:

t

ρ

F

ρ

α

ρ

−



= −

_{ +}



; ; ,

; ;

_{, ,}

_,

1

1

; ;1

;1

;

y x

_xt

_vt

x

xv

v v

t

t

ρ

β

α



− −

−

−

₋

_



−

−

− − +

+

_

(2.5) (iii) On taking

v

→

0

in (2.1), we get

[

]

2 2 0

( ) ( !)

(1

)

(1

)

n

n _{n n}

n

ρ

α

β

∞ =

×

+

+

∑

; ;( ) ( , )

( , ) ; ;( )qp

( , )

a _n

n _{x y n b}

2:0;0;1;1;0 2:0;0;0;1;1

[( ) :1,1,1,0,1],

(1 )

[( ) :1,1,1,0,1],

p p

q

q

a

t

F

b

ρ +

− +



= −





[ :1,1,0,1,1],[

1:1,1,0,0,1]:

[1

:1,1,0,0,1],[ : 0,1,0,1,1,]:

x

w

ρ

µ

α

ρ

− +

+

; ;

; ; ;

,

, ,

,

1

1

; ; ;1

;1

;

x

wt

_{w w}

yt

_wx

w

t

t

λ ρ

β

α



− − − +

− −

−

_

−



−

−

− − − +

+

_

(2.6)

(iv) By writing

μ = 1, λ = 0

, p = q and

a = b

_{j j}, j = 1, 2, …, p and letting

v

→

0

, in (2.1), we have (2.6)

[

]

2 2 0

( ) ( !)

(1

)

(1

)

n

n n n

n

ρ

α

β

∞

=

×

+

+

∑

( , )

( , )

( , )

n n x y n

L

α β

J x w t

α

2:0;0;1;0

2:0;0;1;1

[ :1,1,1,1],

(1 )

[1

:1,1,0,1],

t

ρ

F

ρ

α

−



= −

_{ +}



[ :1,1,0,1]: ; ; ; ;

[ : 0,1,1,1]: ; ;1

;1

;

x

w

ρ

ρ

β

α

− −

−

− − +

+

,

,

,

1

1

wt

_w

yt

_wx

t

t



−

₋



−

−

_

(v) Taking

v

→

0

,

w

→

0

, p = q and

a = b

_{j j}, j = 1, 2, …, p in (2.1), we get

[

]

2

( , ) ( , ) 2

0

( ) ( !)

_{( )}

(1

)

(1

)

n n

n x y n

n n n

n

_L

α β

_{L x t}

α

ρ

α

β

∞

=

+

+

∑

1:0;0;1;0

2:0;0;1;1

[ :1,1,1,1]

:

(1 )

[1

:1,1,0,1],[ : 0,1,1,1]:

t

ρ

F

ρ

α

ρ

−



= −

_{ +}



2

; ; ; ;

, ,

,

; ;1

;1

;1

1

xt

_x

yt

_x

t

t

ρ

β

α

− −

−

−

₋





− − +

+

−

−

_

(2.7) (vi) Taking

β

=

0

, y = 0 in (2.7), we get

[

]

2 ( )

0

( ) ( !)

_{( )}

(1

)

n n

n x n

n n

n

_L

α

_{L x t}

α

ρ

α

∞

=

+

∑

1:0;0;0

2:0;0;1

[ :1,1,1]

:

(1 )

[1

:1,1,1],[ : 0,1,1]:

t

ρ

F

ρ

α

ρ

−



= −

_{ +}



2

; ; ;

, ,

; ;1

α

;1

xt x x

t

− − −

−

₋





− − +

−

_

(2.8)

These are all the bilateral (bilinear) generating relations for the class of generalized hypergeo-metric functions (1.1), whereas the results for the modified Jacobi polynomial, Laguerre poly-nomial of two variables and Laguerre poly-nomial are believed to be new.

Proof of (2.2). The result (2.2) can also be deduced by

using (1.9) and the same techniques as followed in the previous result.

(i). By letting limit

v

→

0

in (2.2), we obtain 2

( , ) ( , ) 0

( !)

_{( ; )}

(1

) (1

_n

)

_n n x y n n

n

n

_L

α β

_φ

_x

_λ

_t

α

β

∞

=

+

+

∑

1 2:0;0;0

2:0;0;1

[1:1,1,1],

(1 ) (1

)

[1

:1,1,1],

x x

t

te

− − −λ

F



_β

= −

−

_{ +}



1 2 3

[ :1,0,1]: ; ; ;

, ,

[1:1,0,1] : ; ;1

;

x

t t t

α

−

− − −





− − +

_

(2.9)

where

1

(

1) ,

2

,

(

)(1 )

y e

t

yt

t

t

e

t

t

e

t

λ

λ λ

−

−

=

=

−

−

−

( )

( )

3

1

.

1 (1 )

yx e

t

t

e

t

λ

λ

−

=

−

−

is the bilateral generating relation for the Laguerre polynomial of two variables, which is believed to be new. Proof of (2.3). The result (2.3) can also be deduced by using (1.10) and the same techniques as followed in the result (2.1).

Application. Applying limit

_v

_→

₀

_{on (2.3), we obtain}

the result

2

( , ) 0

( !)

_{( , ) ( ; )}

(1

) (1

_n

)

_n n n n

n

n

_L

α β

_{x y f x a t}

α

β

∞

=

+

+

∑

1:0;0;0

1:0;0;1

[ :1,0,1]:

(1 )

[1

:1,1,1], :

x axt

x

t

−

e F



_β

= −

_{ +}

−



; ;

;

,

,

1

1

; ;1

;

n

x

_yt

_xyt

ytax

v

t

t

α



− −



−





−

_



₋





₋

_









− − +

_

3. Conclusion

By employing the technique used in the proof of (2.1) and adjusting the parameters one can easily get the bilateral generating relations.

Acknowledgements

The authors are thankful to the referee for giving some useful suggestions to improve this paper.

REFERENCES

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