The research paper published by IJSER journal is about RELATIONSHIP BETWEEN DOUBLE LAPLACE TRANSFORM AND DOUBLE MELLIN TRANSFORM IN TERMS OF GENERALIZED HYPERGEOMETRIC FUNCTION WITH APPLICATIONS 1

ISSN 2229-5518

Relationship between Double Laplace Transform and Double Mellin Transform in Terms of Generalized Hypergeometric Function with Applications

Yashwant Singh’ and Harmendra kumar mandia”

Abstract: The object of this paper is to establish a relation between the double Laplace transform and the double Mellin transform. A double Laplace-Mellin transform of the product of H-functions of one and two variables is then obtained. Application, summation formula and some interesting special cases have also been discussed.

Key words: Double Laplace Transform, Double Mellin Transform, Mellin-Bernes Contour Integral, H-function of Two Variables, Generalized Hypergeometric function.

(2000 Mathematical subject classification: 33C99)

1. INTRODUCTION:

——————————  ——————————

a n z n

If F ( p1 , p2 ) is the Double Laplace transform of

f ( x, y) ,

F a ; b

; z F

P ; z j 1 ,

P Q P Q P Q Q

then

bQ n 0

j 1

j n

F ( p , p )

e p1x p2 y f (x, y)dxdy; Re( p ) 0, Re( p ) 0

(1.3)

1 2 1 2

0 0

If M ( p1 , p2 ) is the Double Mellin transform of then

(1.1)

f ( x, y) ,

Where for brevity, ( a p )denotes the array of parameters

a1 , ..., a p with similar interpretation for ( bq ) etc. . For further details one can refer Rainville [4].

M ( p , p )

x p1 1 y p2 1 f ( x, y)dxdy ; p

0, p 0

The following formula is required in the proof:

1 2 1 2

0 0

(1.2)

xs 1 yt 1H ax

0 0

, by dxdy

Yashwant singh

Department of Mathematics,

a s /

b t /

s t s t

S.M.L. (P.G.) College, Jhunjhunu , Rajasthan, India

E-mail: ysingh23@yahoo.co.in Harmendra Kumar Mandia Department of Mathematics,

Shri JagdishPrasad Jhabermal Tibrewala

, 2 3

(1.4)

University,

Chudela, Jhunjhunu, Rajasthan, India

E-mail: mandiaharmendra@gmail.com

The following theorem is also required in the proof:

Theorem: If F ( p1 , p2 ) is the laplace transform and

M ( p1 , p2 ) is the Mellin transform of

f (t1 , t2 ) , then

The research paper published by IJSER journal is about RELATIONSHIP BETWEEN DOUBLE LAPLACE TRANSFORM AND DOUBLE MELLIN TRANSFORM IN TERMS OF GENERALIZED HYPERGEOMETRIC FUNCTION WITH APPLICATIONS 2

ISSN 2229-5518

( p )s1 (

p )s2

(1.9)

F ( p , p )

1 2 M (s

1, s 1)

1 2 1 2

s1 0 s2 0

s1 !

s2 !

(1.5)

2. A DOUBLE HANKEL TRANSFORM:

1 1 ; ; u v m,n

Provided

f (t1 , t2 ) is continuous for all values of t1 and t2,

x y FS gR

0 0

kS dx y H p,q

the Laplace transform of | f (t1 , t2 ) | exists and the series

on the right hand side of F ( p1 , p2 ) converges.

cx y

1 ( g j ,G j ) p

1 ( h j , H j )q

H ax

, by dxdy

H[ X ]

represents the H-function of Fox [1].

a ( 2 r ) / b ( 2 r ) /

( ) 1 f (r )

The H -function of two variables (Mittal and Gupta [2], p.172) using the following notation, which is due

r 0

m n n ,n m m

1 ( g j ,G j )n , 1 (1 d j

/ /

D j ;

D j )m2

essentially to Srivastava and Panda ([6], p.266, eq. (1.5) et

H p q q

q3 ,q p1 p2 p3

ca b

seq.)is defined and represented as:

2 3 2 3

1 ( h j ,H j )m , 1 (1 c j

C j ;

C j )n2

f r F F g G d

r D D

x 0,n1:m2 ,n2 :m3 ,n3

x (a j ;

j , Aj )1, p1 :(c j , j )1, p2 ,(e j , E j )1, p3

1 j j ,

j ,n 1 ( j ,

j ) p , 1

n2 1

j j , j

H[x, y] H y

H p ,q : p ,q : p ,q

y (b j ; j ,B j )1,q1 :(d j , j )1,q2 ,( f j ,Fj )1,q3 r r

1 e j

E j ,

E j ,m 1 ( h j , H j )q , 1 c j

p3 n2 1

C j , C j

1 ( , ) ( ) ( ) x y d d

r r

1 j j j j j

1 q

= 4 2 2 3

a r r A A

L1 L2

(1.6)

1 j j j j j

1 p1

(2.1)

Where

Provided,

( , )

(1 a j j

j 1

Aj )

, 0; 0; arg c 1 , 0

j n1 1

(a j j

Aj ) (1

j 1

bj j

(1.7)

B j )

Where

f (r)

j 1 r d r

S r !

( ) j 1

(1 c j j

) (d j j )

j 1

j r

j 1

m q n p

2 p2 q2

H H G G

(c j j

) (1

d j j )

j 1 j m 1

j 1 j n 1

j j j j

j n2 1

j m2 1

(1 e j

E j ) ( f j

Fj )

(1.8)

Re[(

r di / Di

hj / H j

] 0;i

1,..., m2 ; j

1,...,m

3 ( ) p q

Re[(

r fi / Fi

hj / H j

] 0;i

1,..., m3; j

1,..., m

j n3 1

(e j

E j ) (1 f j

j m3 1

Fj )

The research paper published by IJSER journal is about RELATIONSHIP BETWEEN DOUBLE LAPLACE TRANSFORM AND DOUBLE MELLIN TRANSFORM IN TERMS OF GENERALIZED HYPERGEOMETRIC FUNCTION WITH APPLICATIONS 3

ISSN 2229-5518

g G d

r s1 D D

1 ci

1 g j

m n n ,n m m

1 ( j ,

/ /

j )n , 1 (1

j j ;
j )m2

Re r

C G

0; i

1,.., n2 ; j

1,..., n

H 2 3 2 3 ca b

p q1 q2

q3 ,q p1 p2 p3

( h , H ) , (1 c

2r s1 C ; C )

i j 1

j j m 1
j j j n2

1 ei

1 g j

1 j 2 j , j

,n 1 ( j ,

j ) p , 1 j
1 j , j

f r s

F F g G d

r s D D

Re r

Ei G j

0; i

1,.., n3 ; j

1,..., n

1 e r s2 E , E

p3 n2 1

, ( h , H ) , 1 c

r s1 C , C

j j j m 1

1 p3

j j q j j j
n2 1 p2

b r s r s B B

R S or R S

1 and | at u v | 1 [none of

1 j 1 j

j j j
q1

a r s r s A A

k j ( j

1, 2, ...S ) is a negative integer or zero].

1 j 1 j

j j j
p1

(3.1)

Proof: To prove (2.1), we use series representation for the

generalized hypergeometric function, substitute the Mellin-

Provided the conditions are same as that of (3.1) with

Bernes contour integral for H [cx y

] on the left hand side

Re(p1)>0, Re(p2)>0.

then interchange the order of contour integral and the

( x, y) -integrals. Finally we arrive at our result on

evaluating the ( x, y) - integral by using the result (1.3).

Proof: In(1.1)put

1 1 u v

3. Application:

f (t1 , t2 )

t1 t2 FS

gR ; kS

; dx y

If f (t1 , t2 )

1 1

1 2 S

gR ; kS

; dxu yv

m,n p ,q

ct1 t2

1 ( g j ,G j ) p

1 ( h j , H j )q

H at

, bt

m,n p ,q

ct1 t2

1 ( g j ,G j ) p

1 ( h j , H j )q

H at

, bt

;

and use (3.1) to get,

r s1 ) / (1

r s2 ) / 1

(1.5) becomes the double Laplace-Mellin transform of the product of H-functions of one and two variables and take the following form:

M ( p1 , p2 ) a b

r 0

( )

( g ,G ) , (1 d

f (r )

r s1 1 D ; D )

P1t1

P2t2 1 1

; ; u v

m n2 n3 ,n m2 m3

p q q q ,q p p p

ca / b /

1 j j n 1

j j j m2

r s 1

e t1 t2 FS gR

kS d x y

1 2 3 1 2 3

( h , H ) , (1 c

1 C ; C )

1 j j m 1

0 0

j j j n2

m,n p ,q

ct1 t2

1 ( g j ,G j ) p

1 ( h j , H j )q

H at1 , bt2

dt1dt2

1 f r s2 1 F , F

, ( g ,G ) , 1 d

r s11 D , D

( P )s1 (

P2 )

( r s1 )/ (

r s2 )/

) 1 ( )

j j j n 1

1 p3

r s2 1

j j p j j j
n2 1 q2

r s1 1

a b ( f r

1 e E , E

, ( h , H ) , 1 c C , C

s1 0 s2

0 r 0 s1 !s2 !

j j j m 1

1 p3

j j q j j j
n2 1 p2

1 b r s1 1

r s2 1 B B

j j j j j

1 q1

1 a r s1 1

r s2 1 A A

j j j j j

1 p1

The research paper published by IJSER journal is about RELATIONSHIP BETWEEN DOUBLE LAPLACE TRANSFORM AND DOUBLE MELLIN TRANSFORM IN TERMS OF GENERALIZED HYPERGEOMETRIC FUNCTION WITH APPLICATIONS 4

ISSN 2229-5518

( P )s1 (

P2 )

( r s )/ (

r s )/ 1

Hence

F ( p1 , p2 )

= the right hand side of (3.1).

s1 0 s2

0 r 0

a 1 b s1 !s2 !

2 ( )

f (r)

( g ,G ) , (1 d

r s1 D ; D )

4. Summation formula:

H m n2 n3 ,n m2 m3

ca / b /

1 j j n 1

j j j m2

p q1 q2

q3 ,q p1 p2 p3

( h , H ) , (1 c

2r s1 C ; C )

P1t1

P2t2 1 1

; ; u v

r s2

1 j j m 1

r s1

j j j n2

e t1 t2 FS gR

0 0

kS d t1 t2

1 f j

Fj ,

Fj ,n 1 ( g j ,G j ) p , 1 d j p3 n2 1

D j , D j

1 e r s2 E , E

, ( h , H ) , 1 c

r s1 C , C

m,n

1 ( g j ,G j ) p

j j j m 1

1 p3

j j q j j j
n2 1 p2

H p ,q

ct1 t2

1 ( h j , H j )q

H at1 , bt2

dt1dt2

1 b r s1

r s2 B B

j j j j j

1 q1

a r s1

r s2 A A

( P )s1 (

s !s

P2 ) a (

r s1 )/ b (

r s2 )/

( ) 1 f (r)

1 j j j j j

1 p1

s1 0 s2

0 r 0 1 2

Provided the conditions are same as that of (3.1) with

( g ,G ) , (1 d

r s1 D ; D )

Re(p1)>0, Re(p2)>0.

H m n2 n3 ,n m2 m3

ca / b /

1 j j n 1

j j j m2

p q1 q2

q3 ,q p1 p2 p3

( h , H ) , (1 c

2r s1 C ; C )

1 j j m 1

j j j n2

5. Special Cases:

1 f r s2 F , F

, ( g ,G ) , 1 d

r s1 D , D

j j j n 1

1 p3

j j p j j j
n2 1 q2

1 e r s2 E , E

, ( h , H ) , 1 c

r s1 C , C

(1)

j j j m 1

1 p3

j j q j j j
n2 1 p2

In (3.1) let P1→0, P2→0 and γ = η = 1, to get the following

b r s r s B B

result:

1 j 1 j

j j j

q1

a r s r s A A

1 j 1 j

j j j

1 1 ; ; u v

1 p1

(4.1)

t1 t2

0 0

m,n

FS gR kS

1 ( g j ,G j ) p

d x y

Evaluating the left hand side of (4.1),using ([5],

H p ,q

ct1 t2

1 ( h j , H j )q

H at1 , bt2

dt1dt2

eq.(8.5.6),p.150), the following summation formula is

obtained:

a ( r s1 )b (

r 0

r s2 ) f (r )

m n n ,n m m

1 ( g j ,G j )n , 1 (1 d j (

r ) D j ;

D j )m2

H p q q

q3 ,q p1 p2 p3

ca b

( h , H ) , (1 c (

r )C ; C )

f (r )a b P1

r 0

ur vr

2 3 2 3

1 j j m 1

j j j n2

m n2

n3 ,n 2 m2 m3

1 ( g j ,G j )n ,(1 u

, ),(1 v

, ),

H p 2 q q

q3 ,q p1 p2 p3

cP1

P2 ab (h ,H ) , (1 c ; C )

1 f (

r ) F , F

, ( g ,G ) , 1 d (

r ) D , D

(1 d ; D ) , 1

f , F

, ( g ,G ) , 1 d , D

1 j j j p3 n 1

j j p n2 1
j j j q2

1 j j m2 1

j j q3 n 1
j j p n2 1
j j q2

1 e (

r ) E j , E j

,m 1 ( h j , H j )q , 1 c j (

p3 n2 1

r )C j , C j

1 e , E

, ( h , H ) , 1 c , C

1 j j p3 m 1

j j q n2 1
j j p2

1 b (

r ) j (

r ) B j j B j

1 b j j B j

1 a (

r ) (

r ) A A

1 q1

1 a j j Aj

1 j j j j j p1

(5.1)

The research paper published by IJSER journal is about RELATIONSHIP BETWEEN DOUBLE LAPLACE TRANSFORM AND DOUBLE MELLIN TRANSFORM IN TERMS OF GENERALIZED HYPERGEOMETRIC FUNCTION WITH APPLICATIONS 5

ISSN 2229-5518

(2) In (4.1) take p1 = q1 = 0, to get the double Laplace-

Hankel transform of the product of three single H-functions of Fox as:

P1t1

P2t2 1 1

; ; u v

e t1 t2 FS gR

0 0

kS d x y

m,n p ,q

ct1 t2

1 ( g j ,G j ) p

1 ( h j , H j )q

H at1 , bt2

dt1dt2

( P )s1 (

P2 )

( r s1 )/ (

r s2 )/

) 1 ( )

s1 0 s2

a b

0 r 0 s1 !s2 !

( f r

( g ,G ) , (1 d

r s1 D ; D )

H m n2 n3 ,n m2 m3

ca / b /

1 j j n 1

j j j m2

p q1 q2

q3 ,q p1 p2 p3

( h , H ) , (1 c

2r s1 C ; C )

1 j j m 1

j j j n2

1 f r s2 F , F

, ( g ,G ) , 1 d

r s1 D , D

j j j n 1

1 p3

j j p j j j
n2 1 q2

1 e r s2 E , E

, ( h , H ) , 1 c

r s1 C , C

j j j m 1

1 p3

j j q j j j
n2 1 p2

(5.2)

Provided the conditions are same as that of (2.1) with p1 =

q1= 0; Re(p1)>0, Re(p2)>0.

REFERENCES:

1. 1. Fox,C. 1961.The G and H-functions as symmetrical Fourier kernels, Trans. Amer. Math. Soc.,98,395-429.

2. 2. Mittal,P.K. and Gupta,K.C. 1972. An integral involving generalized function of two variables, Proc. Indian Acad. Sci. Sect. A. 75, 117-123.

3. 3. Srivastava , H.M. and Panda,R. 1976 a. Some bilateral generating functions for a class of generalized hypergeometric polynomials, J. Reine Angew. Math. 283/284, , 265-274.

4. 4. Nair, V.C. and Samar,M.S. 1975. A relation between the Laplace transform and the Mellin transform with applications, Portugaliae Mathematica, vol.34, Fasc. 3, 149-155.

5. 5. Srivastava,H.M. , Gupta, K.C. and Goyal, S.P. 1982. The H-function

of One and Two Variables with Applications, South Asian Publishers,

New Delhi.