Gen. Math. Notes, Vol. 6, No. 1, September 2011, pp. 61-72
ISSN 2219-7184; Copyright © ICSRS Publication, 2011
www.i-csrs.org
Available free online at http://www.geman.in
The n- Dimensional Generalized Weyl
Fractional Calculus Containing to
n- Dimensional H -Transforms
V.B.L. Chaurasia1 and Ravi Shanker Dubey2
1
2
Department of mathematics, University of Rajasthan
Jaipur-302055, India
E-mail: chaurasiavbl82@gmail.com
Department of mathematics, Yagyavalkya Institute of Technology,
Sitapura, Jaipur -302022, India
E-mail: ravishankerdubey@indiatimes.com
(Received: 24-5-11/Accepted: 8-8-11)
Abstract
The object of this paper is to establish a relation between the n-dimensional H transform involving the Weyl type n- dimensional Saigo operator of fractional integration.
Keywords: Fractional Integral, Riemann-Liouville Operator, Gauss Hypergeometric
function, H − function , Fox’s H-function, G-function.
1
Introduction
Our purpose of this paper is to establish a theorem on n-dimensional H -transforms involving
with Weyl type n-dimensional Saigo operators.
Further, a few interesting and elegant results as special cases of our main results has also
been recorded.
62
2
V.B.L. Chaurasia et al.
Fractional Integrals and Derivative
An interesting and useful generalization of both the Riemann-Liouville and Erdélyi-Kober
fractional integration operators are introduced by Saigo [9], [10] in terms of Gauss’s
hypergeometric function as given below.
Let α, β and η are complex numbers and let y ∈ R + = (0, ∞) . Following [9], [10] the
fractional integral (Re (α ) > 0) and derivative (Re (α ) < 0) of the first kind of a function
f(y) on R + are defined respectively in the following forms
y−α−β y
t
α −1 
I
f=
F  α + β, − η;α;1−  f ( t ) dt; R ( α) > 0
∫ ( y −t)
2 1
Γ ( α) 0
y
0,y
α,β,η
dn α + n,β − n,η − n
I
f,
dyn 0,y
=
0 < R ( α) +n ≤ 1,
( n = 1,2,...) ,
(1)
(2)
F (α , β ; γ ; . ) is Gauss’s hypergeometric function. The fractional integral
(Re (α ) > 0) and derivative (Re (α ) < 0) of the second kind are given by
where
2 1
J
α,β,η
1 ∞
y
−α−β 
f=
(t − y)α−1 t
F  α + β, −η;α;1 − f ( t ) dt, R ( α ) > 0
∫
2 1
y, ∞
Γ ( α) y
t
= ( − 1)n
d n α +n,β −n,η
I
f,
dy n y,∞
0 < R (α ) + n ≤ 1 (n = 1,2,...).
(3)
(4)
The Riemann-Liouville, Weyl and Erdélyi-Kober fractional calculus operators follow as
special cases of the operators I and J as given below
R
α
0,y
α ,−α ,η
f = I
=
W
α
y,∞
0,y
α ,η
0,y
1 y
α −1 f (t) dt ,
∫ (y − t)
Γ(α ) 0
R (α ) > 0
d n α +n
R
f , 0 < R (α ) + n ≤ 1, (n = 1,2,...)
dy n 0,y
f = J
α ,−α ,η
y,∞
= (−1) n
E
f =
0,y
1 ∞
α −1 f (t) dt, R (α ) > 0
∫ (t − y)
Γ(α ) y
α +n
dn
W
f , 0 < R (α ) + n ≤ 1, (n = 1,2,...)
n
y,∞
dy
α ,0,η
f = I
f =
f =
y −α −η y
α −1 tη f (t) dt, R(α ) > 0,
∫ (y − t)
Γ(α ) 0
(5)
(6)
(7)
(8)
(9)
The n- Dimensional Generalized Weyl…
K
α ,η
y,∞
f = J
α ,0,η
y,∞
f =
63
yη ∞
α −1 t −α −η f (t) dt, R(α ) > 0.
∫ y (t − y)
Γ(α )
(10)
Following Miller [8, p.82], we denote by u1 the class of functions f (x1) on R+ which are
−ξ
infinitely differentiable with partial derivatives of any order behaving as 0 (|x1 | 1 )
when x1 → ∞ for all ξ . Similarly by u2, we denote the class of functions f ( x1 , x 2
1
)
on R+
× R+, which are infinitely differentiable with partial derivatives of any order behaving as
−ξ
−ξ
0(|x1 | 1 |x 2 | 2 ) when x1 → ∞, x2 → ∞ for all ξi (i = 1, 2) .
On the same way, we denote the class of functions f ( x1 , x 2 ,...,x n ) on R+ ×… × R+, which
are infinitely differentiable with partial derivatives of any order behaving as
−ξ
−ξ
ξ
0 (| x1 | 1 |x 2 | 2 ...|x n | n ) when xi → ∞, where i = 1, 2,..., n for all ξi (i = 1, 2,..., n) by un.
The n-dimensional operator of Weyl type fractional
Re (α i ) > 0, where i = 1, 2,..., n is defined in the class un by,
integration
of
orders
n
 αi ,βi ,γ i 
 pi β i 
J
[f
(p
,p
,...,p
)]
=
 p ,∞ 
∏
∏


1
2
n
i
i =1 
i =1
 Γ(αi ) 

n
∞ ∞ ∞ n 
p 
α −1 −α −β 
...∫ ∏(ti −pi ) i ti i i F αi +βi , −γi ; αi ; 1− i f ( t1,t2,...,tn ) dt1dt2...dtn,
∫
2 1
p1 p2 pn i=1 
ti 
.∫
(11)
where β i and γ i , i = 1, 2, …, n are real numbers.
More generally, the operator (11) of Weyl type fractional calculus in n-variables is defined by
the differ-integral expression as,
n
∑r
 αi ,βi ,γ i 
 pi

i =1
 J p ,∞ [f (p1 ,p 2 ,...,p n )] = ∏ 
∏
 (−1)
α
Γ
(
+r
)
i
i =1 
i
1
=
i
i 


n
n
βi
n
∂ r +r +...+r
∂p1r ∂p2r ... ∂pn r
1
1
2
n
2
n
∞ ∞ ∞ n 
p 
α −1 −α −β 
...∫ ∏(ti −pi ) i ti i i F αi +βi , −γi ; αi ; 1− i f ( t1,t2,...,tn ) dt1dt2...dtn,
∫
2 1
p1 p2 pn i=1 
ti 
.∫
for arbitrary real (complex) α i and r1 , r2 ..., rn = 0,1, 2,... .
(12)
64
In
V.B.L. Chaurasia et al.
particular,
if
R (α i ) < 0
and
ri
are
positive
integers
such
that
R (α i ) + ri > 0, wherei = 1, 2,..., n, then (12) yields the partial fractional derivative of
f (p1 ,p 2 ,...,p n ).
On the other hand if we set β i = 0 , (12) yields the Weyl type Erdélyi-Kober operators in ndimensions
n  α ,0,γ 
i
i
 αi ,γ i 
K
[f
(p
,p
,...,p
)]=
J
[f (p1 ,p2 ,...,pn )]

∏
∏
1
2
n
 pi ,∞ 
pi ,∞ 
i =1 
i =1 
n
n
n
=∏
i =1
∞
 ∞ ∞
. ∫
...∫
∫
pn
 p1 p2
3
ri
∑
 pi

∂ r1 +r2 +...+rn
i =1
(
1)
−


∂p1r1 ∂p2 r2 ... ∂pn rn
 Γ(α i +ri ) 
βi

αi +ri −1 −αi −γ i 
t
t
f
,
,...,
dt
dt
...dt
(
−
p
)
t
t
t
(
)
.
i
i
i
1
2
n
1
2
n



n
∏
i =1
(13)
n-Dimensional Laplace and H -Transactions
The Laplace transform ζ (pi) of a function f (xi) ∈ un is defined as
ζ (p1 ,p2 ,...,pn ) = L[ f (x1 ,x 2 ,...,x n ) ; p1 ,p2 ,...,pn ]
=
∞
∞
∫ ∫ ...∫
0
0
−
∞
e
n
∑ pi x i
i =1
f (x1 ,x 2 ,...,x n ) dx1dx 2 ...dx n
(14)
0
where R( pi ) > 0, wherei = 1, 2,..., n .Similarly, the Laplace transform of
f [u1
x12 − λ12 H (x1 − λ1 ), u 2
x 22 − λ22 H (x 2 − λ2 ),..., u n
x n2 − λn2 H (x n − λn ) ],
is defined by the Laplace transform of F (x1 ,x 2 ,...,x n ) where
F (x1,x2,...,xn) = f [u1 x12 −λ12 H(x1 −λ1),u2 x22 −λ22 H(x2 −λ2),...,un x2n −λn2 H(xn −λn)],
x i > λi > 0, where i = 1, 2,..., n;
(15)
and H (t) denotes Heaviside’s unit step function.
Definition: By n-dimensional H -transform ϕ ( p1 , p 2,..., p n ) of a function F ( x1 , x 2 ,...,x n ) , we
mean the following repeated integral involving n-different H -functions
The n- Dimensional Generalized Weyl…
65
M1,N1;M2,N2;...;Mn,Nn
ϕ ( p1,p2,..., pn ) = ϕ
[F( x1, x2 ,...,xn ) ; a1, a2 ,..., an ; p1,p2,..., pn ]
P1,Q1;P2,Q2 ;...;Pn,Qn
∞ ∞ ∞
= ∫ ∫ ...∫
λ1 λ2 λn


(a ,α ;A ) ,(a ,α )

M ,N 
1,N ij ij N +1,P 
ij
ij
ij
k

i i
i
i i
a −1
i
. (pi xi ) i . H
(pi xi )

∏
P ,Q 
β
β
(b
,
)
,(b
,
;B
)
i =1 
i i
ij ij 1,Mi ij ij ij Mi+1,Qi 



(16)
. F (x1 ,x 2 ,..., x n ) dx1dx 2 ...dx n ,
n
here we suppose that λi > 0, k i > 0, where i = 1, 2,..., n; ϕ ( p1 , p 2,..., p n ) , exists and belongs to un.
Further suppose that,
k
1
| arg p i | < T π ,
2 i
(17)
M
N
Q
P
i
i
i
i
T = ∑ | β | + ∑ A a − ∑ |B β | −
α > 0,
∑
i j = 1 ij
j = 1 ij ij j=M +1 ij ij j = N +1 ij
i
i
(18)
where,
where i=1,2,…,n.
The H -function appearing in (16), introduced by Inayat-Hussain ( [1], see also [14]) in
terms of Mellin-Barnes type contour integral, is defined by,
H
M ,N
P ,Q
z

( a j ,α j ; A j )1 ,N , ( a j ,α j ) N + 1 , P
( b j , β j )1 ,M , ( b j , β j ; B j ) M + 1 , Q
 = 1

2π i
∫
i∞
− i∞
ψ (ξ ) z ξ d ξ ,
(19)
where
M
ψ (ξ ) =
∏
j=1
Q
∏
j= M +1
N
Γ(b j − β jξ ) ∏ {Γ(1 − a j + α jξ )}
Aj
j=1
{ Γ(1 − b + β ξ ) } ∏
Bj
j
j
P
j= N +1
,
(20)
Γ(a j − α jξ )
which contains fractional powers of some of the Γ-functions. Here and throughout the paper
a j ( j =1,…, P ) and b j ( j = 1,…, Q )
are
complex
parameters.
α j ≥ 0 ( j = 1,…, P ) , β j ≥ 0 ( j = 1,…, Q ) , (not all zero simultaneously) and the exponents
A j ( j = 1,…, N ) and B j ( j = M + 1,…, Q ) can take on non-integer values. The contour in
(19) is imaginary axis R(ξ ) = 0 . It is suitably indented in order to avoid the singularities
of the Γ-functions and to keep these singularities on appropriate sides. Again, for
A j ( j = 1,…, N ) not an integer, the poles of the Γ-functions of the numerator in (16) are
converted to branch points. However, as long as there is no coincidence of poles from any
66
V.B.L. Chaurasia et al.
Γ (b j − β jξ ), (j = 1,...,M) and Γ (1 − a j + α jξ ), ( j = 1, … , N ) pair the branch cuts can be
chosen so that the path of integration can be distorted in the usual manner.
For the sake of brevity
T =
M
∑
j=1
4
|βj | +
N
∑
j=1
A jα j −
Q
∑
j= M +1
|B j β j | −
P
∑
j= N +1
α j > 0.
(21)
Relationship between n-Dimensional H -Transforms in
Terms of n-Dimensional Saigo Operator of Weyl Type
To prove the theorem in this section, we need the following n-dimensional H -transform
φ1 ( p1 , p 2,..., p n ) of F ( x1 , x2,..., xn ) defined by,
M1+1,N1;M2+1,N2;... ;Mn+1,Nn
φ1 ( p1,p2,...,pn ) = H
[F(x1, x2,..., xn );a1, a2,..., an; p1,p2,...,pn ]
P1+1,Q1+1;P2+1,Q2+1;...;Pn+1,Qn+1
{
∞ ∞
∞ n
α −1
...∫ ∏ (pi x i ) i
∫
λ1 λ2 λn i =1

(a ,δ ;A ) ,(a ,δ )
,(1−a ,k ),(α +β +γ −a +1,k ) 
M +2,N 
1,N ij ij N +1,P
ij
ij
ij
i i i i i i i 
k
i
i
i
i i

i
.H
(pi xi )

P +2,Q +2 
(β −α +1, k ),(γ −α +1, k ),(b , β )1,M ,(b , β , B )M +1,Q 
i
i
i i i i i i ij ij
i ij ij ij i i
=∫


.F (x1 ,x 2 ,..., x n ) dx1dx 2 ...dx n ,
(22)
where it is assumed that φ1 ( p1 , p 2,..., p n ) exists and belongs to un as well as ki > 0, where
i = 1, 2, … , n and other conditions on the parameters, in which additional parameters
α i , βi , γ i where i = 1, 2,..., n included correspond to those in (11).
Theorem
1
Let
φ1 ( p1 , p 2,..., p n )
be
given
by
definition
(14)
then
for R (α i ) > 0,λi >0,k i > 0, where i = 1, 2,..., n there holds the formula
n
∏
i =1
 αi ,βi ,γ i 
J p ,∞  [φ (p1 ,p 2 ,...,p n )]=φ1 (p1 ,p 2 ,...,p n )
 i

provided that φ1 (p1 ,p 2 ,...,pn ) exists and belong to un.
Proof: Let R(α i ) > 0, where i = 1, 2,..., n then in view of (11) and (18), we find that
(23)
The n- Dimensional Generalized Weyl…
67
n
 αi ,βi ,γ i 
 pi β i 
J
[
φ
(p
,p
,...,p
)]
=
.
 p ,∞ 
∏
∏


1
2
n
i
i =1 
i =1  Γ(α i ) 

n
∞ ∞ ∞n 
p 
α −1 −α −β 
...∫ ∏(ti −pi ) i ti i i F αi + βi , −γi ; αi ; 1− i  φ ( t1, t2,..., tn ) dt1dt2...dtn ,
∫
2 1
p1 p2 pn i=1 
ti 
.∫
or
 pi βi  ∞ ∞ ∞ n 

p 
αi −1 −αi −βi
= ∏. 
ti
F αi + βi , − γ i ; αi ; 1− i 
 .∫p ∫p ...∫p ∏(ti − pi )
2 1
ti 
1
2
n i =1 
i =1  Γ(αi ) 
n
∞ ∞ ∞ n

. ∫ ∫ ...∫ ∏
 λ1 λ2 λn i=1


(a ,δ ;A )1,N ,(a ,α )N +1,P 
ij ij ij i ij ij i i 
k

a −1 Mi,Ni 
i
(
t
x
)
( xi ti ) i .H
 i i

P ,Q 
(b , β )
,(b , β ;B )

i i
ij ij 1,M ij ij ij M +1,Q 
i
i
i 


.F (x1 ,x 2 ,..., x n ) dx1dx 2 ...dx n } dt1dt 2 ...dt n .
(24)
On interchanging the order of integration which is permissible and on evaluating the
ti where i = 1, 2,..., n integrals through the integral formula

(a ,α ;A ) ,(a ,α )

∞ −µ−ν
x  M,N  k j j j 1,N j j N+1,P 

ν
1
−
(u − x)
F τ ,ω;ν ;1− .H
(au)
du
∫x u
2 1
u  P,Q 
(b , β )1,M,(b , β )M+1,Q 
j j
j j



(a ,α ;A ) ,(a ,α )
,(µ+ν −τ ,k), (µ+ν −ω,k) 
Γ(ν ) M+2,N  k j j j 1,N j j N+1,P
,
(ax)
= µ H


P+2,Q+2
(µ ,k), (µ+ν −τ −ω , k),(b , β )1,M,(b , β )M+1,Q
x
j j
j j

where, R (ν ) > 0,

R  µ +ν +


(25)

k (1 − a ) 
j 
α
j


>0

k (1 − a ) 
j 

R µ +ν − τ − ω +
> 0,| arg z | <


α
j 

1
T π (T is given in (21))
2
(23) can be established by means of the following formula [2, p.399].
1 γ −1
Γ(γ ) Γ( ρ ) Γ(γ + ρ − α − β )
(1 − x) ρ −1 F (α , β ; γ ; x) dx =
∫ x
21
Γ(γ + ρ − α ) Γ(γ + ρ − β )
0
(26)
68
V.B.L. Chaurasia et al.
for R(γ ) > 0, R( ρ ) > 0, R(γ + ρ − α − β ) > 0.
by using the formula, left hand side of (24) becomes
∞ ∞ ∞
=∫
∫ ...∫
λ1 λ2 λn
a −1

n
∏ (t x ) i
i =1

i i
(a , δ ;A )
,(a , δ )
,(1−a ,k ),(α +β +γ −a +1, k )  
M +2,N 
i i i i i i
i 
k ij ij ij 1,Ni j j Ni+1,Pi
i
i
i
.H
(
t
x
)
 i i

P +2,Q +2 
(β −a +1, k ),(γ −a +1, k ),(b , β )1,M ,(b , β )M +1,Q

i
i
i i
i i i
i
ij ij
ij
ij
i
i
i
 

.F(x1 ,x 2 ,..., x n ) dx1dx 2 ...dx n .
M1+2,N1 ;M2+2,N2 ;... ;Mn+2,Nn
=H
[F (x1 , x2,..., xn ) ; α1 ,α 2 ,...,α n ; p1 , p2,..., pn ]
P1+2,Q1+2;P2+2,Q2+2 ;...;Pn+2,Qn+2
= ϕ (p1 ,p2 ,...,p n ).
1
n
As far as the n- dimensional Weyl type operators
 αi ,βi ,γ i 
 preserves the class un, it
 i

∏  J p ,∞
i =1
follows that ϕ (p1 ,p 2 ,...,p n ) also belongs to un.
1
It is interesting to note that the statement of Theorem 1 can easily be extended for arbitrary
real α i where i = 1, 2,..., n by using the definition (12) for the generalized Weyl type fractional
calculus operators and differentiating under the signs of the integrals.
5
Interesting Special Cases
Putting γ i = 0 where i = 1, 2,..., n in theorem 1, we can easily prove Theorem 1(a).
Theorem
1(a).
For
R (α i ) > 0, βi > 0, ri > 0; where i = 1, 2,..., n and
also
let
ϕ (p1 ,p2 ,...,pn ) be given by (14) then there holds the following formula,
n
 αi ,βi ,0 
 [ϕ (p1 ,p 2 ,...,p n )] = ϕ2 (p1 ,p 2 ,...,p n )
 i

∏ .  J p ,∞
i =1
(27)
provided that ϕ2 (p1 ,p 2 ,...,p n ) exists and belongs to un where ϕ 2 is represented by the
repeated integral,
∞ ∞
∞
ϕ2 (p1 ,p2 ,...,p n ) = ∫ ∫ ...∫
λ1 λ2 λn
n

i =1

a −1
∏ (p x ) i
i i
The n- Dimensional Generalized Weyl…
M+1,N 
k
i i
i
.H
(px
i i)
P+1,Q+1
i i

69
(a ,δ ;A )1,N ,(a ,δ )N+1,P ,(α +β −a+1,k ) 
ij ij ij i ij ij i i i i i i 
F(x1,x2,...,xn)dxdx
1 2...dxn.
(β −a+1,k ),(b ,β )1,M,(b ,β , B )M+1,Q 
i i i ij ij
ij ij ij
i
i i 
(28)
For Aij=1, the H -function reduces to Fox’s H-function [5], [6] and then Theorem 1 (a)
reduces to,
n
 αi ,βi ,0 
 [ϕ (p1 ,p 2 ,...,pn )] = ϕ3 (p1 ,p 2 ,...,p n ),
i


∏ J p ,∞
i =1
(29)
provided that ϕ (p1 ,p2 ,...,p n ) exists and belongs to un, where ϕ3 is represented by the
3
repeated integral,


(a ,δ ),(α +β −a +1,k ) 
k iPi iPi i i i i 
ai−1 Mi+1,Ni 
∞ ∞ ∞ n 
i
(px
ϕ3(p1,p2,...,pn) = ∫ ∫ ...∫ ∏ (pi xi ) .H

i i)

P +1,Q +1 
(β −a +1, k ),(b , β ) 
λ1 λ2 λn i=1 
i i 
i
i
i
iQ
iQ
i
i


.F (x1 ,x 2 ,..., x n ) dx1dx 2 ...dx n .
(30)
On employing the identity


M,N  (aP,1) 
M,N
H
x
=G
P,Q  (b ,1) 
P,Q
Q 


a ,...,aP 
x 1
,

b1,..., bQ 


(31)
we see that the n- dimensional H-transform reduces to the corresponding n- dimensional Gtransform θ (p1 ,p 2 ,...,pn ) defined by
M1,N1;M2,N2;... ;Mn,Nn
θ(p1,p2 ,...,pn ) = G
[F (x1, x2,..., xn ) ; α1,α2 ,...,αn ; p1,p2,...,pn ]
P1,Q1;P2,Q2;...;Pn,Qn

k
αi−1 M,N
∞ ∞ ∞
i i
i
= ∫ ∫ ...∫ (px
G
(px
i i)
i i)
P,Q 
λ1 λ2 λn
i i

a ,...,aP 
1i
i
.F (x1,x2,...,xn) dxdx
1 2...dxn,
b ,...,b 
1i Qi

(32)
provided that θ (p1 ,p 2 ,...,pn ) exists and belongs to class un, where ki, are positive
integers, λi > 0, Pi ≤ Qi ,
70
V.B.L. Chaurasia et al.
k
1
| arg p i | < T*π ,
2 i
(33)
T* = 2N + 2M − P − Q ,
i
i
i i
i
(34)
with
where i=1,2,…,n, G
M,N
P,Q
[.] , appealing in (31) and (32) represents Meijer’s G-function
whose detailed account is available from the monograph of Mathai and Saxena [4].
Thus, we obtain the following Theorem 1 (b).
Theorem 1(b). For R (α i ) > 0, βi > 0, ki > 0; where i = 1, 2,..., n being positive integers
and also let θ (p1 ,p 2 ,...,pn ) be given by (31) then the following formula
 αi ,βi ,0 
 [θ (p1 ,p2 ,...,p n )] = θ1(p1 ,p 2 ,...,p n )
i


n
∏ . J p ,∞
i =1
(35)
holds, provided that θ (p1 ,p 2 ,...,p n ) exists and belongs to class un for other conditions on the
1
parameters, in which additional parameters α i , β i and γ i where i = 1, 2,..., n
correspond to those in (32). Here
−α  ∞ ∞ ∞
i ∫ ∫ ...∫

i

 λ λ
1 2 λn
θ1(p1 ,p 2 ,...,p n ) = ∏  k
n
i =1
M +2,N
i
i
.G
P +2,N
i
i
n
a −1

∏ (p x ) i
i =1

i
i

a ,...,aP ,∆ (k ,1−a ), ∆(k , α +β +γ −a +1)  
i
i
i i i i i 
k 1i

i

i
(px)


∆(k , β −a +1), ∆ (k , γ −a +1),bQ ,...,bQ  

i i i
i i i
i
i  

.F (x1 ,x 2 ,..., x n ) dx1dx 2 ...dx n ,
and the symbol ∆ (n,α) represents the sequence of parameters
α α +1
n
,
n
,...,
included
α + n −1
n
On taking γ i = 0, wherei = 1, 2,..., n, (36) becomes
(36)
The n- Dimensional Generalized Weyl…
n
 αi ,βi ,0 
 [θ (p1 ,p2 ,...,p n )] = θ2 (p1 ,p 2 ,...,pn )
i


∏ . J p ,∞
i =1
71
(37)
provided θ (p1 ,p 2 ,...,p n ) exists and belongs to class un, where θ2 is represented by the
2
integral
n
−α
a −1
∞ ∞ ∞ n 
θ 2 (p1 ,p 2 ,...,p n ) = ∏  k i  ∫ ∫ ...∫ ∏ (pi x i ) i
 i  λ λ
i =1
1 2 λn i =1 

a ,...,aP , ∆(k ,α +β −a +1)  
i i i i 
k 1i

i

i
(px)


∆(k ,β −a +1),bQ ,...,bQ  

i i i
i
i  

.F (x1 ,x 2 ,..., x n ) dx1dx 2 ...dx n .
M +1,N
i
i
.G
P +1,N
i
i
6
(38)
Special Case
(i)
Converting our Theorem 1, 1(a) and 1(b) for i=1,2,3; we find the known result defined
by Chaurasia and jain [19]
(ii)
Converting our Theorem 1, 1(a) and 1(b) for i=1,2; we find the known result defined
by Chaurasia and Shrivastava [18],if we tack N = N ′ = 0.
(iii)
Taking Aj = Bj = 1, then Theorem 1, 1(a) and 1(b) for i=1,2; we find the known result
defined by Saigo, Saxena and Ram [13].
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