ISSN 2319-8133 (Online)
(An International Research Journal), www.compmath-journal.org
On Certain Integrals Involving the Product of Multivariable H- Function & M- Series
Priyanka Gupta and Neelam Pandey Department of Mathematics,
Model Science College Rewa – 486003, M.P. INDIA.
(Received on: May 31, 2018) ABSTRACT
Integral formulas (finite and infinite) plays an important role in the study of generalized hypergeometric function. There are a number of paper on such results. In the present paper, we establish certain definite integrals of involving the product of multivariable H-function & M-series. The results are in general character and besides of this have been put in a compact form avoiding the occurrence of infinite series and thus making them useful in applications. Several other new and known results can also be obtained from our main theorems.
Mathematics Subject Classification - 33C20, 33C60, 26A33.
Keywords: Generalized hypergeometric function and fractional calculus, M-series, Special function, Multivariable H-function.
1. INTRODUCTION AND PRELIMINARIES
The multivariable H-function which was introduced and investigated by Srivastava &
Panda12, Eqn. (4.1) in term of a multiple Mellin-Bernes type contour integral as H [z1,...,zr]
j j jr p j j p jr jr pr
qr r j r q j q j
r j j r j
c c
a
d d
b z
z
, 1 ) ( ) ( , 1 1 1 1 , 1 ) ( 1
, 1 ) ( ) ( ,1 1 1 1 1 , 1 ) 1 ( 1 r r 1 1
r r 1 1
,
;...;
, : ,...,
;
,
;...;
, : ,...,
; n
, m
;...;
n , m : n 0,
q , p
;...;
q , p : q
Hp,
r
i
L r
r
i
i i i L r
r ... ,..., z d ...d ,
) 2 (
1
1 1
1
1
(1)
Where 1;and
) 2 )( 1
( )
(
1
,..., ()
1 1
) ( 1 1
) ( 1 1
1
i i j r i j q
j i i j r i j p
n j
i i j r i j n
j
r a b
a
(2)
1
1
1 () ()
1,..., ;) ( ) ( 1
) ( ) ( 1 )
( ) (
1 i r
d c
d c
i i j i j q
m i j i j i j p
n j
i i j i j m i j i j i j n
j i
i i
i i
i
i i
(3)
And
L j
Lj represents the contours which start at the point j and terminate at the pointsj with j ,j1,...,r.
In case r = 2, (1) reduce to the H-function of two variables.
For a detailed definition and convergence conditions of the multivariable H-function, the reader is referred to the original paper by Srivastava and Panda13, we have
1,...,
1 ...
max1 0 ,1
j
r j e
e z z
z O z z
H r (4)
Where
).
..., , 1 ) (
min Re(()
) (
1 d i r
e i
j i j m
i j
i
(5)
For n = p = q = 0 the multivariable H-function breaks up into product of 'r' H-functions and consequently there holds the following result:
r
i
c d c
c
d d
z z
pi i j i j
qi i j i j pr
r j r p j j j
qr r j r q j r j
z
1
, , n
, m
q , p ,
;...;
,
,
;...;
, n
, m
;...;
n , :m 0,0
q , p
;...;
q , :p 0,0
, 1 ) ( ) (
, 1 ) ( ) ( i i
i i ,
1 ) ( ) ( , 1 1 1 1
, 1 ) ( ) ( ,1 1 1 1 1 1 r r 1 1
r r 1
1 H
H , (6)
Where Hm,p,qn()is the familiar H- function.
The well known M-series, which is a particular case of H -function introduced by Inayat Hussain9 and is defined by means of the following series expansion:
0 1 Q
p 1
Q 1 p 1
Q (B) ...(B ) ( 1)
) A ( ...
) A ) (
; B ,..., B
; A ,..., A ( pM
r
r
r r
r r
r
(7)
Provided that C,R() 0, (Aj)r (Bj)r are pochammer symbols.
2. RESULT REQUIRED IN THE SEQUEL
For a0;b0;c4ab0;()1/20the following formulas is defined by Qureshi et al.3, Eqn, (3.1) – (3.3):
). 1 (
2) ( 1
) 4 (
2 2
1 1
0
2
c ab a dx x c
ax b (8)
For a0;b0;c4ab0;()1/20,
). 1 (
2) ( 1
) 4 ( 2 1
2 1 1
0
2
2
c ab b dx x c
ax b x
(9)
For a0;b0;c4ab0;()1/20,
) . 1 (
2) ( 1
) 4
( 2
1 1
0
2
2 2
2
c ab dx x c
a b x
a b (10)
The following formulas9, will also be required in our investigation.
, )
y
; 2b,2c (2a, F )
1 (
0 1
2 c - b
a
r r ry a
y (11)
and
0 1
2 1
2 .
2 , c 1
r) ) (c,
2; c 1 b, - c a, - (c F ) 2; c 1 b, (a, F
r
r
ar
r
(12)
3. MAIN RESULTS
Theorem 3.1. Let
), ..., , 1 ( 0 ) (
2 0, ) 1 ( 0, 0, 0, 4ab c 0, b 0,
a i ei i r
2 ) 1 2 (
1
a b c and c
x ax b
X
2
the following formula holds:
0 2 1 2 1 1
-
- ;X)
2 c 1 b, - c a, - (c F X) 2; c 1 b, (a, F X
yX
H
Z X Zr X r
MQ (AP);(BQ); ) 1 1,...,
0 0
2 /
1 (4 )
) 1 ( 2,
1 ) , ( ) 4 (
1 )
4 (
2 r r r k k
c k ab
M a
r c
r c c
ab c
ab
a
) 13 (
H 1,
) ( ) ( ,1 1 1 1 , 1 ) ( 1 1
, 1 ) ( ) ( ,1 1 1 1 , 1
1 ) ( 1 1 1 1
1 1 1
,
;...;
, : ,...,
; , ,..., 2 ;
1
,
;...;
, : ,...,
; ,
,...,
; ) 4 (
) 4 ( , ...;
; n , m : 1 n 0,
, ...;
; ,q p : 1 1,q p
pr
r j r p j j p j r j j j r
qr r j r q j j j q r
r j j j r r r
r r r
c c
a k
r
d d
k r b
c ab
Z
c ab
Z n
m q p
(13) where ei is defined in (5).
Proof: By virtue of equation (1 ), (7 ), (8 ) and ( 12 ), we have the following:
yX HZ X ZrX rdX
0X 2F1(a,b,c21;X) 2F1(c-a,c-b, c21;X)PMQ(AP);(BQ); ) 1 1,..., 1
- -
0 0 1
1 0
1 - -
) 1 ( ) (
) ( 2,
1 ) , X (
r k
k k
j Q
j
k k j p r j
r X
k B
y X A
a r c
r
c
r
i i
L r
r
i
i i i L r
r Z X d d
,..., ,...,
) ...
2 (
1
1 1
1 1
0
1 - q - p - r -
0 1
1 1
0
dX, X
) ( ,...,
,..., ) ...
2 (
1 2,
1 ) , (
1
r
i
L k
r r
i
i i i L r
r r
r Z d d M k
a r c
r c
where,
) 1 ( ) (
) ) (
(
1 1
k B
y k A
M
k j Q
j
k k j P j
and , ;
1
r
i i
q i
k
p
0 1
1 1
0
) ( ,...,
,..., ) ...
2 (
1 2,
1 ) , (
1 L rk
r
i
i i i L r
r r
r Z d d M k
a r c
r c
r
i
) 1
(
) 2 / 1 (
) 4 (
2 1/2 r p q
q p r c
ab
a r p q
r p
r k
r r
c k ab M a r c
r c c ab c
ab
a (2 )
1 ) 4 ( ) 1 ( 2,
1 ) , ( ) 4 (
1 )
4 (
2 0 0
2 /
1
) 1
(
) 2 / 1 (
) 4 ( ,..., 1 ,...,
... 1
1 1
1 r p q
q p r c
d ab d
Z q
L r
r
i
i i i
L r r
i
k
k r
r ar M k ab c
r c
r c c ab c
ab
a
) 4 ( ) 1 ( 2,
1 ) , ( ) 4 (
1 )
4 (
2 1/2 0 0
) 1
(
) 2 / 1 ,..., (
) 4 ,..., (
) ...
2 (
1
1 1
1
1 r p q
q p d r
c d ab
Z
L r
r
i
i i i L r
r
r
i
i
k
k r
r r
c k ab
M a
r c
r c c
ab c
ab
a
) 4 ( ) 1 ( 2,
1 ) , ( ) 4 (
1 )
4 (
2 0 0
2 / 1
(14) .
H 1,
) ( ) ( ,1 1 1 1 , 1 ) ( 1 1
, 1 ) ( ) ( ,1 1 1 1 , 1
1 ) ( 1 1 1
1 1
1 1
,
;...;
, : ,...,
; , ,..., 2 ;
1
,
;...;
, : ,...,
; ,
,...,
; ) 4 (
) 4 ( , ...;
; n , m : 1 n 0,
, ...;
; q , p : 1 q 1, p
pr
r j r p j j p j r j j j r
qr r j r q j j j q r
r j j j r r r
r r r
c c
a k
r
d d
k r b
c ab
Z
c ab
Z n
m q p
(14)
