Vol 7, No 5 (2016)

Available online throug

ISSN 2229 – 5046

CERTAIN INTEGRAL TRANSFORMATIONS PERTAINING

TO THE MULTIVARIABLE A- FUNCTION

PINKY LATA, MUKESH AGNIHOTRI*

Mathematics, Jaipur, Rajasthan, India.

(Received On: 04-02-16; Revised & Accepted On: 19-05-16)

ABSTRACT

T

he object of the present paper is to obtain few general multiple integral transformations of the multivariable

A-function (1981), as a kernel product with Fox’s H-A-function [3, p. 408] and Laguerre polynomials respectively with the general class of polynomial ([4] and [7]). Several possible cases are also included.

Key Words And Phrases: A-function, H-function, Laguerre polynomials, General class of polynomials.

1. INTRODUCTION

Gautam and Goyal [1981] defined the multivariable A-function, which is a generalization of multivariable H-function of Srivastava and Panda [1976 b]. The definition of multivariable A-function runs as follows:





















=

r r r

r

r r

C r

j r

j C

j j C r

j j

j

r j r j j

j r

j j

j

r C C

C

r

_b

_B

_B

_d

_D

_d

_D

C

C

A

A

a

z

z

A

z

z

F

, 1 ,

1 ' ' , 1 ) ( '

, 1 ,

1 ' ' , 1 ) ( '

1 , ;...; , : ,

, ;...; , : ,

1

₍

_;

_;...;

₎

_;

₍

_,

₎

_;...;

₍

_,

₎

)

,

(

;...;

)

,

(

;

)

;...;

;

(

]

..,

,

[

1 1 1

1 1 1

υ υ

υ λ

µ λ µ λ µ

υ υ

υ

τ

τ

1 1

1 1 1 1 1

1

...

( )... ( )

( ,..., )

...

...

,

(2

)

r

r

s s

r r r r r

r

L L

s

s

s

s z

z ds

ds

θ

θ

πω

=

_{∫ ∫}

Φ

(1.1)

,

1

where

ω

= −

1 1

1 1

1 1

( ) ( ) ( ) ( )

1 1

1 1

( ) ( ) ( ) ( )

1 1

(

)

(1

)

( )

,

1,..., ,

(1

)

(

)

i i i i

j j i j j i

j j

C D

i i i i

j j i j j i

j j

d

D s

C s

s

i

r

d

D s

C s

µ λ

µ λ

τ

τ

= =

= + = +

Γ

−

Γ −

+

=

∀ =

Γ −

+

Γ

−

∏

∏

∏

∏

(1.2)

( ) ( ) ( )

1 1

1 1

1

( ) ( ) ( )

1 1

1 1

(1

)

(

)

( ,...,

)

,

(1

)

(

)

r r

i i i

j j i j j i

i i

j j

r C r D r

i i i

j j i j j i

i i

j j

a

A s

b

B s

s

s

b

B s

a

A s

µ λ

µ λ

= =

= =

= =

= + = +

Γ −

+

Γ

−

Φ

=

Γ − +

Γ

−

∑

∑

∏

∏

∑

∑

∏

∏

(1.3)

Here

µ λ υ

, , , ,

C

µ λ υ

_i

, ,

_{i i}

and c

_i are non-negative integers and all

a s b s d

_j

' ,

_j

' ,

( )_ji

' ,

s

τ

( )_ji

' ,

s B

( )_ji

'

s

complex numbers.

The multiple integral defining the A-function of r-variables converges absolutely if

*

0

i

ξ

=

, (1.4)

0

i

d

>

, (1.5)

and

arg( )

2

i

z

k i

π

ξ

<

η

, (1.6)

© 2016, IJMA. All Rights Reserved 108 where

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

1 1 1 1

{

}

{

}

{

}

{

}

i i

i i i i

j j j j

c

D C

A B D C

i i i i

i j j j j

j j j j

A

B

D

C

ν

ξ

− −

= = = =

=

∏

∏

∏

∏

, (1.7)

[ ]

* ( ) ( ) ( ) ( )

1 1 1 1

0,1

i i

c C

i i i i

i j j j j

j j j j

img

A

B

D

C

υ υ

ξ

= = = =





=

_

−

+

−

_



∑

∑

∑

∑



, (1.8)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1 1 1 1

Re

,

i ci i i

C

i i i i i i i i

i j j j j j j j j

j j j j j j j j

A

A

B

B

D

D

C

C

µ λ υ

µ

λ υ

η

= = = = = = = =





=

_

−

+

−

+

−

+

−

_



∑

∑

∑

∑

∑

∑

∑

∑



∀ =

i

1,..., .

r

(1.9)

If we take

A s B s C

_j

' ,

_j

' ,

( )_ji

'

s and D

_j( )i

'

s

as real and

µ

=

0

, the A-function reduces to multivariable H-function of Srivastava and Panda [1976 b].

Srivastava [4]introduce the general class of polynomials (see also Srivastava and Singh [7])

, 0

(

)

[ ]

,

0,1, 2,...,

!

k k

k k

S

z

A

z

k

β α

β α

α β

β

_β

     

=

−

=

∑

=

(1.10)

where

α

an arbitrary positive integer and coefficients

A

_β,k

( ,

β

k

≥

0)

are arbitrary constants, real or complex.

2. THE MAIN RESULTS

(i) 1 1 1 1 1

1 1 1 1 1

0 0

...

...

r

(

...

r

)

[ (

...

r

) ]

h

r r r r r

x

σ

x

σ

k x

ρ

k x

ρ σ

S

_αβ

η

k x

ρ

k x

ρ ∞ ∞

− −

_{+ +}

_{+ +}

∫ ∫

1 1 1

1 1

1 1

1 1 , : , ;...; ,

,0

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

1 1

( ,

),..., (

,

)

(

...

)

.

...

...

(

,

),..., (

,

)

r r r

r r

p p

m

p q r r C c c r

q q

r r

z X

e

e

H

k x

k x

A

dx

dx

g

g

z X

ρ ρ µ λ µ λ µ λ

υ υ υ

ε

ε

ξ

γ

γ









_

_

+ +





_

_









_

_

_

_

r r

r r

m r

c c

p C q r hk k k k

k r

s

A

A

k

k

k

µ

λ

µ

λ

µ

λ

υ

υ

υ

β

α

β

α

_η

_ξ

β

ξ

, : , ;...; ,

, ;...; , : ,

, 0

1

1 1

1 1

!

)

(

)

,...,

(

+ +

+ + + + −

   

=

−

_Ψ

_∑

−

=

[

]

( )

1 _1,

1,

1 1 1 _1,

1

: '

,...,

, 1

(

) ;

,...,

:

1

:

,...,

,

1

(

) ;

,...,

:

r

j j j j j j _r j j j r j _q

r r j j j r j _p

g

S

hk

N

N

S

N

n

N

n

e

S

hk

N

N

ρ σ ξ σ

ξ

σ

γ

γ

γ

σ

ε

ε

ε



−





_

−

−

+



_









− +

−

−

_

− −

+

_

1 1

' ( ) ' ' ( ) ( ) 1

1, 1, 1,

' ( ) ' ' ( ) ( )

1, 1, 1,

( ;

;...;

) ( ,

)

;...; (

,

)

...

( ;

;...;

)

(

,

)

;...; (

,

)

r

r

r r r

j j j j j j j

r r r

j j j C j j c j j c

r

Z

a A

A

C

C

b B

B

d D

d

D

Z

υ

τ

υ

τ

υ









, (2.1)

where

( ) ( )

1 1

1

...

(

1 1

...

)

i i

i

r r

i r r r

X

=

x

ξ

x

ξ

k x

ρ

+ +

k x

ρ η , (2.2)

1 1

...

r r

S

σ

σ

σ

ρ

ρ

= +

+ +

, (2.3)

1 1

1

1 1 1

( ,...,

)

( ...

)

,...,

r r

r r r

k

k

k

k

σ σ

ρ ρ

σ σ

− − −

Ψ

=

, (2.4)

( ) ( )

1 1

...

i i

r

i i

r

N

n

ξ

ξ

ρ

ρ

= +

+ +

, (2.5)

and

( ) ( ) 1

1 1

1 1

...

i i

r r N

i i r

Z

z

k

k

ξ ξ

ρ ρ

ξ

− − −

The above integral formula (2.1) is valid under the following sufficient conditions:

(a)

k

_i

>

0,

ρ

_i

>

0,

n

_i

≥

0,

ξ

_i( )i

>

0,

∀

i j

,

∈

{1,..., }

r

, (2.7) (b)

Re(

σ

_i

)

>

0,

i

=

1,...,

r

and

1

min

Re( )

Re

1

r

j i i

i j

g

S

N

i

m

δ

γ

=













> −

−





_



_



≤ ≤ 











∑

, (2.8)

where

( ) ( )

min Re

,

1,...,

i j

i i i

j

d

j

D

δ

=











_





_







=

µ













, (2.9)

(c) m, n, q are integers such that

1

≤ ≤

m

q and p

≥

0,

ε

_j

>

0 (

j

=

1,..., ),

p

1 1

1 1 1 1 1

0(

1,..., ),

0,

0

p q m q p

j j j j j j

j j j j m j

j

q

γ

ε

γ

γ

γ

ε

= = = = + =

>

=

Ω ≡

∑

−

∑

<

Ω ≡

∑

−

∑

−

∑

>

and

arg( )

1

₂

2

ξ

<

π

Ω

(2.9)

(d)

A

_β,kare arbitrary constants, real or complex and β,k≥0.

(e) Conditions corresponding appropriately (1.4) through (1.6) are satisfied by each of the multivariable A-function occurring in (2.1). Here Hm_{p q,},0 _{ }z denotes the familier H-function of C. Fox ([3], p. 408, see also [5], p. 310).

(ii) 1 1 1 1 1

1 1 1 1 1

0 0

...

...

r

(

...

r

)

[ (

...

r

) ]

h

r r r r r

x

σ

x

σ

k x

ρ

k x

ρ σ

S

_αβ

η

k x

ρ

k x

ρ ∞ ∞

− −

_{+ +}

_{+ +}

∫ ∫

1

1 1 1 1

1 1

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

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

exp

(

...

r

) .

(

...

r

)

r r

...

...

r r

r u

r r w r r C c c r

r r

z x

k x

k x

L

k x

k x

A

dx

dx

z x

ξ

ρ ρ ρ ρ µ λ µ λ µ λ

υ υ υ

ξ

γ

γ











−

+ +





+ +



_

_

















1 1

1 ,

0 1 1

,

2 :

,

1;...;

,

1

(

)

( 1)

( ,...,

)

.

2,

1:

1, ;...;

1,

( )!

!

w S

r r

k hk k

r k

k r r

k

k

A

A

C

c

c

w

k

β α

α β

µ λ

µ λ

µ λ

β

γ

_{η γ}

υ

υ

υ

 

−  

−

=

+

+

+

−

−

=

Ψ

+

+

+

+

∑

[

] [

]

[

]

' ( )

1 1 1 1 1,

1 1

1

;

,...,

, 1

;

,...,

, ( ;

;...;

) ;

1

;

,...,

,

r

r r r r j j j

r r

S

hk

S

hk

u

a A

A

S

hk

u

w

υ

ξ ρ

ξ ρ

ξ ρ

ξ ρ

ξ ρ

ξ ρ

 − −

− −

+



_{− −}

_{+ +}



1

1

' ' ( ) ( ) 1

1 1 1 1 1, 1,

' ( ) ' ' ( ) ( )

1, 1, 1,

(1

;

), ( ,

)

;...; (1

;

); (

,

)

...

( ;

;...;

)

(

,

)

;...; (

,

)

r

r

r r

j j r r r r j j

r r r

j j j C j j c j j c

r

C

C

b B

B

d D

d

D

υ υ

ζ

σ ρ ξ ρ τ

σ ρ ξ ρ

τ

ζ



−

−

_





, (2.10)

where ( )u

( )

w

L

z

be the Laguerre polynomial of order u and degree w in z,

1

0,

0,

0,

0, Re(

)

0,

1,..., , Re( )

, Re( )

0,

r i i

i i i i

i i

w

k

ρ

ξ

σ

i

r

S

ξ δ

γ

ρ

=





≥

>

>

>

> ∀ =

> −

_

_

>





∑

(2.11)

1

( ,...,

k

k

r

)

Ψ

, S and

δ

_i being given by (2.4), (2.3) and (2.8), respectively,

( )

i i i

z

i

k

i

ξ ρ

ζ

=

γ

− , i=1,…,r and conditions given by (1.4) through (1.6) are assumed to hold the multivariable A-function.

3. PROOFS

To prove the main results, we take some assumptions for convenience

∑

n s

_{i i} and

∑

ξ

_j( )i

s

_idenotes the r-terms sums

1

r

i i i

n s

=

∑

and ( )

1

r i

i i

i

s

ξ

=

∑

, respectively

∀ =

j

1,..., .

r

(3.1) Also, let

1

1 1 1 1

1 1

1 1

1 1 , : , ;...; ,

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

0 0

...

...

r

(

...

r

)

r r

...

...

,

r r

r

r r r C c c r

r r

z x

x

x

f k x

k x

A

dx

dx

z x

ξ

σ σ ρ ρ µ λ µ λ µ λ

υ υ υ

ξ ∞ ∞

− −









∆ =

+ +

_

_









∫ ∫

© 2016, IJMA. All Rights Reserved 110 where the

X

_i are defined by (2.2) and the function f is such that the multiple integral converges. On replacing the multivariable A-function occurring in (3.2) by contour integral given by (1.1), under the various conditions stated with (2.1), we find that

1 1

1 1 1 1

1

...

( )... ( )

( ,...,

)

...

(2

)

r

r

s s

r r r r

r

L L

s

s

s

s z

z

θ

θ

πω

∆ =

∫ ∫

Φ

( ) ( )

1 1 1 1 1 1

1 1 1

0 0

...

isi

...

r r isi

(

...

r

)

n si i

r r r

x

σ ξ

x

σ ξ

k x

ρ

k x

ρ

∞ ∞

+ − + −



_∑

_∑

_∑

+ +





∫ ∫

}

1

1 1 1 1

(

...

r

)

...

...

.

r r r r

f k x

ρ

+ +

k x

ρ

dx

dx

ds

ds

(3.3)

Now we integrate the innermost

( ,...,

x

₁

x

_r

)

-integral by using the following form of a known result [1, p. 173].

1 1 1 1 1

1 1 1 1 1 1

0 0

...

...

r

(

...

r

)

(

...

r

)

...

r r r r r r

x

σ

x

σ

k x

ρ

k x

ρ σ

f k x

ρ

k x

ρ

dx

dx

∞ ∞

− −

_{+ +}

_{+ +}

∫ ∫

(

) (

)

(

1 1

)

1 1 ... 1

1

1 1 0

...

( ,...,

)

( )

,

...

r r

r r

r

r r

k

k

σ ρ

σ ρ

z

σ ρ σ ρ σ

f z dz

σ ρ

σ ρ

∞

+ + + −

Γ

Γ

Ψ

Γ

+ +

∫

(3.4) where

Ψ

( ,...,

k

₁

k

_r

)

is given by (2.4) and

min

{

,

, Re(

)

}

0

1

≤ ≤

i

r

k

i

ρ

i

σ

j

>

then (3.3) reduces in the following form

(

) (

)

(

)

1 1

* *

1 1

1

1 1 1 1 _{* *}

1 1

...

( ,...,

)

...

( )... ( )

( ,...,

)

...

(2

)

...

r

r

r r

s s

r

r r r r

r

L L r r

k

k

s

s

s

s Y

Y

σ ρ

σ ρ

θ

θ

πω

σ ρ

σ ρ

Γ

Γ

Ψ

∆ =

Φ

Γ

+ +

∫ ∫

1 1 ... 1

1 0

( )

...

,

r r

r

z

σ ρ σ ρ σ

f z dz ds

ds

∞

+ + + −











∫



(3.5)

where

Ψ

( ,...,

k

₁

k

_r

),

N

_i and S are given by (2.4), (2.5) and (2.3) respectively, and

( )

,

i j

j

i i

Y

zk

ξ ρ

∑

=

(3.6)

* ( )

1

,

1,..., .

r

i

i j j j

i

s

j

r

σ

σ

ξ

=

=

+

∑

∀ =

(3.7)

Now in the integral (3.5), we set

1 1 ,0

,

1 1

( , ),..., (

,

)

( )

,

( ,

),..., (

,

)

p p

m h

p q

q q

e

e

f z

z H

z

S

z

g

g

σ α

β

ε

ε

ξ

_γ

_γ

γ









=

_

_

_

_









(3.8)

and evaluate the z-integral by following familiar formula (when n=0), expressing the Mellin transform of Fox’s H-function [5, p.311, eq (3.3)]

{

,

}

1 1

,

1 1

(

)

(1

)

(

) :

.

(1

)

(

)

m n

j j j j

j j

m n s

p q q p

j j j j

j m j n

B s

A s

M H

zx

s

z

B s

A s

β

α

β

α

= = −

= + = +

Γ

+

Γ −

−

=

Γ −

−

+

∏

∏

∏

∏

(3.9)

Interpret the resulting

( ,..., )

s

₁

s

_r - integral as an A-function of r-variable, we will obtain the required result given in (2.1).

Moreover to establish the other main integral (2.10) we can find relationship (3.5) in similar way and then we set

( )

exp(

)

h

f z

=

z

σ

−

γ

z S

_βα



_

z

η



_

Evaluate the innermost z-integral by using to a slightly modified version of following well-known integral [2, p. 292, eq. (1)]

{

( )

}

(

1) ( )

(

);

.

! (

1)

x _s

m

s

m

s

M e

L

x s

m

s

γ _α

α

γ

γ

α

−

Γ − + + Γ

₋

=

Γ − +

(3.11)

If we interpret the resulting multiple contour integral as an A-function of r-variable, we will get desired result (2.10).

4. SPECIAL CASES

(1) For the general class of polynomials, we take the case of Hermite polynomials ([8, p. 106, eq. (5.54)] and [7, p.

158]) by setting 2

[ ]

2

1

2

S

z

z

H

z

β

β β





=

_

_





in which case

2,

,

( 1)

k k

A

_β

α

=

= −

.

(i)

Integral 1(a): The result (2.1) reduces in following form

1 1

1

1 1 2 2

1 1 1

0 0 _{1 1}

1

...

...

(

...

)

2

(

...

)

r r

r h

r r r _h

r r

x

x

k x

k x

H

k x

k x

β β

σ

σ σ ρ ρ

β _{ρ ρ}

η

η

∞ ∞

+

− −

_{+ +}



_



_



+ +







∫ ∫

1 1 1

1 1

1 1

1 1 , : , ;...; ,

,0

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

1 1

( ,

),..., (

,

)

(

...

)

.

...

...

(

,

),..., (

,

)

r r r

r r

p p

m

p q r r C c c r

q q

r r

z X

e

e

H

k x

k x

A

dx

dx

g

g

z X

ρ ρ µ λ µ λ µ λ

υ υ υ

ε

ε

ξ

_γ

_γ









_

_

+ +





_

_









_

_

_

_

2

1 1 1

0 1 1

,

:

,

;...;

,

( )!( 1)

( ,...,

)

.

,

1:

, ;...;

,

(

2 )!( )!

k

r r

s k hk

r

k r r

r

m

k

k

A

r

q C

p

c

c

k

k

β

µ λ

µ λ

µ λ

β

ξ

η ξ

υ

υ

υ

β

     

− −

=

+ +

−

=

Ψ

+ +

+ +

−

∑

[

]

( )

1 _1,

1,

1 1 1 _1,

1

: '

,...,

, 1

(

) ;

,...,

:

1

:

,...,

,

1

(

) ;

,...,

:

r

j j j j j j _r j j j r j _q

r r j j j r j _p

g

S

hk

N

N

S

N

n

N

n

e

S

hk

N

N

ρ σ ξ σ

ξ

σ

γ

γ

γ

σ

ε

ε

ε



_

−



_



_

−

−

+



_





_{− +}

₋

₋

_

_{− −}

₊

_







1

1

' ( ) ' ' ( ) ( ) 1

1, 1, 1,

' ( ) ' ' ( ) ( )

1, 1, 1,

( ;

;...;

) ( ,

)

;...; (

,

)

...

( ;

;...;

)

(

,

)

;...; (

,

)

r

r

r r r

j j j j j j j

r r r

j j j C j j c j j c

r

Z

a A

A

C

C

b B

B

d D

d

D

Z

υ

τ

υ

τ

υ









(4.1)

Valid under the same conditions as obtainable from (2.1).

(ii)

Integral 1 (b): The result (2.10) reduces in following form

1 1

1

1 1 _{2 2}

1 1 1

0 0 _{1 1}

1

...

...

(

...

)

2

(

...

)

r r

r h

r r r _h

r r

x

x

k x

k x

H

k x

k x

β β

σ

σ σ ρ ρ

β _{ρ ρ}

η

η

∞ ∞

+

− −

_{+ +}



_



_



+ +







∫ ∫

1

1 1 1 1

1 1

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

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

exp

(

...

r

) .

(

...

r

)

r r

...

...

r r

r u

r r w r r C c c r

r r

z x

k x

k x

L

k x

k x

A

dx

dx

z x

ξ

ρ ρ ρ ρ µ λ µ λ µ λ

υ υ υ

ξ

γ

γ











−

+ +





+ +



_

_

















2

1 1 1

0 1 1

,

2 :

,

1;...;

,

1

( 1)

( )!( 1)

( ,...,

)

.

2,

1:

1, ;...;

1,

( )!

(

2 )!( )!

w S k

r r

k hk r

k r r

k

k

A

C

c

c

w

k

k

β

µ λ

µ λ

µ λ

γ

β

_{η γ}

υ

υ

υ

β

 

−  

−

=

+

+

+

−

−

=

Ψ

+

+

+

+

−

∑

[

] [

]

[

]

' ( )

1 1 1 1 1,

1 1

1

;

,...,

, 1

;

,...,

, ( ;

;...;

) ;

1

;

,...,

,

r

r r r r j j j

r r

S

hk

S

hk

u

a A

A

S

hk

u

w

υ

ξ ρ

ξ ρ

ξ ρ

ξ ρ

ξ ρ

ξ ρ

 − −

− −

+



_{− −}

_{+ +}



1

1

' ' ( ) ( ) 1

1 1 1 1 1, 1,

' ( ) ' ' ( ) ( )

1, 1, 1,

(1

;

), ( ,

)

;...; (1

;

); (

,

)

...

( ;

;...;

)

(

,

)

;...; (

,

)

r

r

r r

j j r r r r j j

r r r

j j j C j j c j j c

r

C

C

b B

B

d D

d

D

υ υ

ζ

σ ρ ξ ρ τ

σ ρ ξ ρ

τ

ζ



−

−

_





© 2016, IJMA. All Rights Reserved 112 Valid under the same conditions as obtainable from (2.10).

(2) If we set

,

1

1

,

(

1)

k

k

and A

_β

β ν

α

β

ν

+





=

_{= }

_

+





the general class of polynomial reduces in Laguerre polynomials ([8, p. 106,

eq. (15, 16)] and [7, p. 159]) where Laguerre polynomials is given by

( )

0

(

)

[ ]

.

( )!

k

k

z

L

z

k

k

β ν

β

β ν

β

=

+



 −

=

_

_

−





∑

(i)

Integral 2(a): The result (2.1) reduces in following form

1 1 1 1 ( ) 1

1 1 1 1 1

0 0

...

...

r

(

...

r

)

[ (

...

r

) ]

h

r r r r r

x

σ

x

σ

k x

ρ

k x

ρ σ

L

_βν

η

k x

ρ

k x

ρ ∞ ∞

− −

_{+ +}

_{+ +}

∫ ∫

1 1 1

1 1

1 1

1 1 , : , ;...; ,

,0

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

1 1

( ,

),..., (

,

)

(

...

)

.

...

...

(

,

),..., (

,

)

r r r

r r

p p

m

p q r r C c c r

q q

r r

z X

e

e

H

k x

k x

A

dx

dx

g

g

z X

ρ ρ µ λ µ λ µ λ

υ υ υ

ε

ε

ξ

γ

γ









_

_

+ +





_

_









_

_

_

_

1 1 1

0 1 1

,

:

,

;...;

,

(

)

( ,...,

)

.

,

1:

, ;...;

,

( )!

k

r r

s hk

r

k r r

r

m

k

k

A

r

q C

p

c

c

k

k

β

_{β ν}

_η

µ λ

µ λ

µ λ

ξ

ξ

υ

υ

υ

β

− −

=

+ +

+



 −

=

Ψ

_

_

+ +

+ +

−





∑

[

]

( )

1 _1,

1,

1 1 1 _1,

1

: '

,...,

, 1

(

) ;

,...,

:

1

:

,...,

,

1

(

) ;

,...,

:

r

j j j j j j _r j j j r j _q

r r j j j r j _p

g

S

hk

N

N

S

N

n

N

n

e

S

hk

N

N

ρ σ ξ σ

ξ

σ

γ

γ

γ

σ

ε

ε

ε



_

−



_



_

−

−

+



_





_{− +}

₋

₋

_

_{− −}

₊

_







1

1

' ( ) ' ' ( ) ( ) 1

1, 1, 1,

' ( ) ' ' ( ) ( )

1, 1, 1,

( ;

;...;

) ( ,

)

;...; (

,

)

...

( ;

;...;

)

(

,

)

;...; (

,

)

r

r

r r r

j j j j j j j

r r r

j j j C j j c j j c

r

Z

a A

A

C

C

b B

B

d D

d

D

Z

υ

τ

υ

τ

υ









(4.3)

Valid under the same conditions as obtainable from (2.1).

(ii)

Integral 2 (b): The result (2.10) reduces in following form

1 1 1 1 ( ) 1

1 1 1 1 1

0 0

...

...

r

(

...

r

)

[ (

...

r

) ]

h

r r r r r

x

σ

x

σ

k x

ρ

k x

ρ σ

L

_βν

η

k x

ρ

k x

ρ ∞ ∞

− −

_{+ +}

_{+ +}

∫ ∫

1

1 1 1 1

1 1

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

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

exp

(

...

r

) .

(

...

r

)

r r

...

...

r r

r u

r r w r r C c c r

r r

z x

k x

k x

L

k x

k x

A

dx

dx

z x

ξ

ρ ρ ρ ρ µ λ µ λ µ λ

υ υ υ

ξ

γ

γ











−

+ +





+ +



_

_

















1 1 1

0 1 1

,

2 :

,

1;...;

,

1

(

)

( ,...,

)

.

2,

1:

1, ;...;

1,

( )!

k

r r

hk r

k r r

k

k

A

C

c

c

k

k

β

β ν

µ λ

µ λ

µ λ

η γ

υ

υ

υ

β

−

=

+

+

+

+



 −

Ψ

_

_

+

+

+

+

−





∑

[

] [

]

[

]

' ( )

1 1 1 1 1,

1 1

1

;

,...,

, 1

;

,...,

, ( ;

;...;

) ;

1

;

,...,

,

r

r r r r j j j

r r

S

hk

S

hk

u

a A

A

S

hk

u

w

υ

ξ ρ

ξ ρ

ξ ρ

ξ ρ

ξ ρ

ξ ρ

 − −

− −

+



_{− −}

_{+ +}



1

1

' ' ( ) ( ) 1

1 1 1 1 1, 1,

' ( ) ' ' ( ) ( )

1, 1, 1,

(1

;

), ( ,

)

;...; (1

;

); (

,

)

...

( ;

;...;

)

(

,

)

;...; (

,

)

r

r

r r

j j r r r r j j

r r r

j j j C j j c j j c

r

C

C

b B

B

d D

d

D

υ υ

ζ

σ ρ ξ ρ τ

σ ρ ξ ρ

τ

ζ



−

−

_





(4.4)

Valid under the same conditions as obtainable from (2.10).

(2) For the Jacobi polynomials [8, p. 68, eq. (15, 16)] and [7, p. 159] by setting

1 ( , )

[ ]

s t

[1 2 ]

S z

_β

=

P

_β

−

z

in which case

α

=

1

and _,

(

1)

(

1)

k k

k

s

s t

A

s

β

β

β

β

+



 + + +

= 



₊

(i)

Integral 3(a): The result (2.1) reduces in following form

1 1 1 1 ( , ) 1

1 1 1 1 1

0 0

...

...

r

(

...

r

)

s t

[1 2 (

...

r

) ]

h

r r r r r

x

σ

x

σ

k x

ρ

k x

ρ σ

P

_β

η

k x

ρ

k x

ρ ∞ ∞

− −

_{+ +}

₋

_{+ +}

∫ ∫

1 1 1

1 1

1 1

1 1 , : , ;...; ,

,0

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

1 1

( ,

),..., (

,

)

(

...

)

.

...

...

(

,

),..., (

,

)

r r r

r r

p p

m

p q r r C c c r

q q

r r

z X

e

e

H

k x

k x

A

dx

dx

g

g

z X

ρ ρ µ λ µ λ µ λ

υ υ υ

ε

ε

ξ

_γ

_γ









_

_

+ +





_

_









_

_

_

_

₁ 1 1

0 1 1

,

:

, ;...;

,

( ,...,

)

.

,

1:

, ;...;

,

r r

s hk

r

k r r

r

m

s

t

k

s

k

k

A

r

q C

p

c

c

k

k

β

β

β

µ λ

µ λ

µ λ

ξ

ξ

υ

υ

υ

β

− −

=

+ +

+

+ + +







=

Ψ

_

_

_

+ +

+ +

−







∑

[

]

( )

1 _1,

1,

1 1 1 _1,

1

: '

,...,

, 1

(

) ;

,...,

:

1

:

,...,

,

1

(

) ;

,...,

:

r

j j j j j j _r j j j r j _q

r r j j j r j _p

g

S

hk

N

N

S

N

n

N

n

e

S

hk

N

N

ρ σ ξ σ

ξ

σ

γ

γ

γ

σ

ε

ε

ε



_

−



_



_

−

−

+



_





_{− +}

₋

₋

_

_{− −}

₊

_







1 1

' ( ) ' ' ( ) ( ) 1

1, 1, 1,

' ( ) ' ' ( ) ( )

1, 1, 1,

( ;

;...;

) ( ,

)

;...; (

,

)

...

( ;

;...;

)

(

,

)

;...; (

,

)

r

r

r r r

j j j j j j j

r r r

j j j C j j c j j c

r

Z

a A

A

C

C

b B

B

d D

d

D

Z

υ

τ

υ

τ

υ









(4.5)

Valid under the same conditions as obtainable from (2.1).

(ii)

Integral 3 (b): The result (2.10) reduces in following form

1 1 1 1 ( , ) 1

1 1 1 1 1

0 0

...

...

r

(

...

r

)

s t

[1 2 (

...

r

) ]

h

r r r r r

x

σ

x

σ

k x

ρ

k x

ρ σ

P

_β

η

k x

ρ

k x

ρ ∞ ∞

− −

_{+ +}

₋

_{+ +}

∫ ∫

1

1 1 1 1

1 1

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

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

exp

(

...

r

) .

(

...

r

)

r r

...

...

r r

r u

r r w r r C c c r

r r

z x

k x

k x

L

k x

k x

A

dx

dx

z x

ξ

ρ ρ ρ ρ µ λ µ λ µ λ

υ υ υ

ξ

γ

γ











−

+ +





+ +



_

_

















1 1 1

0 1 1

,

2 :

,

1;...;

,

1

( 1)

( ,...,

)

.

2,

1:

1, ;...;

1,

( )!

w S

r r

hk r

k r r

s

t

k

s

k

k

A

C

c

c

k

k

w

β

_β

_β

_{µ λ}

_{µ λ}

_{µ λ}

γ

_γ

υ

υ

υ

β

−

−

=

+

+

+

+

+ + +







−

=

Ψ

_

_

_

+

+

+

+

−







∑

[

] [

]

[

]

' ( )

1 1 1 1 1,

1 1

1

;

,...,

, 1

;

,...,

, ( ;

;...;

) ;

1

;

,...,

,

r

r r r r j j j

r r

S

hk

S

hk

u

a A

A

S

hk

u

w

υ

ξ ρ

ξ ρ

ξ ρ

ξ ρ

ξ ρ

ξ ρ

 − −

− −

+



_{− −}

_{+ +}



1

1

' ' ( ) ( ) 1

1 1 1 1 1, 1,

' ( ) ' ' ( ) ( )

1, 1, 1,

(1

;

), ( ,

)

;...; (1

;

); (

,

)

...

( ;

;...;

)

(

,

)

;...; (

,

)

r

r

r r

j j r r r r j j

r r r

j j j C j j c j j c

r

C

C

b B

B

d D

d

D

υ υ

ζ

σ ρ ξ ρ τ

σ ρ ξ ρ

τ

ζ



−

−

_





(4.6)

Valid under the same conditions as obtainable from (2.10).

(2) If we take

β

→

0

and

µ

=

0

in result (2.1), we obtain a known result obtained by Srivastava and Panda [6, p. 354, eq. (1.8)].

(3) If we take

β

→

0

and

µ

=

0

in result (2.10), we obtain a known result obtained by Srivastava and Panda [6, p. 354, eq. (1.14)].

ACKNOWLEDGEMENT

© 2016, IJMA. All Rights Reserved 114

REFERENCES

1. L. K. Bhagchandani, Some Triple Whittaker Transformation of Certain by Hyper geometric Function Rev. Mat. Hisp.- Amer. (4)37, 129-146 (1977).

2. A. Erdelyi, W. Magnus, F. Obehettinger and F. G. Tricomi- Tables of Integral Transforms, Vol. II McGraw Hill Book Co., New York, Toronto and London, 1954.

3. C. Fox, The G and H- function as Symmetrical Fourier Kernels, Trans. Amer. Math. Soc. 98, 395-429 (1961). 4. H. M. Srivastava, A Contour Integral Involving Fox’s H-function, Indian J. Maths. 14, 1-6 (1972)

5. H. M. Srivastava and R. Panda, Some Operations Techniques in the Theory of Special Function, Nederl. Akad. Wetensch. Proc. Ser. A76=Indag. Math. 35, 308-319 (1973).

6. H. M. Srivastava and R. Panda, Some Multiple Integral Transformations Involving the H- Funtions of Several Variables, Nederl. Akad. Wetensch. Proc. Ser. A, 82 (3) 353-362, (1979).

7. H. M. Srivastava and N. P. Singh, The Integration of Certain products o f the Multivariable H-Function with a General Class of Polynomials, Rend. Cic. Mat. Palermo (2) 32, 157-187 (1983).

8. G. Szego, Orthogonal Polynomials, Amer. Math. Soc. Collog. Publ. 23, Fourth Edition, amer, math. Soc. Providence, Rhode Island (1975)

Source of support: Nil, Conflict of interest: None Declared