Journal of the Applied Mathematics, Statistics and Informatics (JAMSI), 7 (2011), No. 2
NEW FINITE DOUBLE INTEGRAL FORMULAE
INVOLVING POLYNOMIALS AND FUNCTIONS
OF GENERAL NATURE
PRAVEEN AGARWAL AND SHILPI JAIN
________________________________________________________________________
Abstract
The aim of the present paper is to establish two new finite double integral formulae involving product of the Hfunction of two variables, general sequence of function and the general class of polynomials. These double
integral formulae are unified in nature and act as key formulae from which we can obtain as their special cases,
double integral formulae concerning a large number of simpler special functions and polynomials. For the sake
of illustration, we record here eight special cases of our main formulae which are also new and of interest by
themselves. The findings of the present results are basic in nature and are likely to find useful applications in
several fields.
Mathematics Subject Classification 2000: 33B15, 33C45, 33C60, 33C70
Additional Key Words and Phrases: H-function of two variables, general sequence of function, general class
of polynomials.
________________________________________________________________________
1.
INTRODUCTION
The general sequence of function
R
(α , β )
n
[x] occurring in this paper is a special case of
the sequence of functions given by Agarwal and Chaubey [1, p.1155] see also [11, p.447,
problem (16)] and will be defined and represented in the following manner:
γ n x kn  cx q + d δ n k ne sx r
b
(α , β ) 
− sx r 
R n  x; a, b, c, d ; p, q; γ , δ ; e  =






k′
n
∑
m, v, u , t , e
( −1)t + m ( −v )u ( −t )e
q 
( −α − γ n )e
pe + rm + λ + qu   cx

(α )t s m
( −β − δ n )v 

k
(1 − α − t )e

n  cx q + d 


m !u !v !t !e !
v
 ax p 


 b 


v
( x )rm
(1.1)
where
∞
n
v
n
t
∑ ≡ ∑ ∑ ∑ ∑ ∑ ,
m,v,u,t ,e m = 0 v = 0 u = 0 t = 0 e = 0
(1.2)
and the infinite series on the right hand side of (1.1) is absolutely convergent.
The following explicit series form of R (nα , β ) [x] for s=0 [3,
p.544, Eq. (1.2)] will be required in the derivation of our main formulas:
P. AGARWAL, S. JAIN
R
=
(α , β )
n
[x; a, b, c, d ; p, q; γ , δ ;1]
R′
∑ θ 1∗ ( v, u , t , e ) x
v, u , t , e
−v + δ n
 c q
1
+
,
x


 d

(1.3)
where
γ n k n d δ n − v −1 t −v −t
( ) ( )u ( )e
b
θ ∗ ( v, u , t , e ) =
1
a
 
(α )t ( c )v  
b
pt ( −α − γ n )
e
(1 − α − t )e
v !u !t !e !k ′
n
pe + λ + qu 
 .
k

n
( − β − δ n )v 
(1.4)
R ′ = kn + qv + pt ,
(1.5)
and
n
v
n
t
∑ ≡ ∑ ∑ ∑ ∑ .
v,u ,t ,e v = 0 u = 0 t = 0 e = 0
(1.6)
It may be pointed out that
R
(α , β )
n
[x] unifies and extends a large number of
named classical polynomials and polynomials studied by several research workers.
The general class of polynomials introduced by Srivastava [ 8, p.1, Eq. (1)]:
Sn [ x ] =
m
[n / m] (− n)
k
mk A
, (n=0, 1, 2…)
∑
x
,
n
k
k!
k =0
where m is arbitrary positive integers and the coefficients A
(1.7)
are
arbitrary
(n, k ≥ 0)
n,k
constants, real or complex. On suitable specializing the coefficients A
, m [ x ] yields
n, k S n
a number of known polynomials as its special cases. These include, among others, the
Hermite polynomials, the Jacobin polynomials, the Laguerre polynomials, the Bessel’s
polynomials and several others [10, pp.158-161].
The H-function of several variables is defined and represented as follows [9, pp.251-252,
Eqn’s (C.1)-(C.3)]:
NEW FINITE DOUBLE INTEGRAL FORMULAE INVOLVING POLYNOMIALS
AND FUNCTIONS OF GENERAL NATURE
(r)
(r) (r)
z
0,n:m1 , n1 ;...; m r , n r  1 (a j ; α ′j ,..., α j )1, p :(c′j , γ ′j )1, p1 ;...; (c j , γ j )1, pr
H [z1 ,...,z r ] ≡ H
M
p,q : p1 , q1 ;...; p r , q r  z (b j ; β ′j ,..., β (j r ) )1,q :(d ′j , δ ′j )1, q ;...; ( f j( r ) , F j( r ) )1,q
1
r
 r
 1 
=

 2π i 
r
ξ
ξ
∫ L ∫ φ (ξ )... φ (ξ )ψ (ξ ,...ξ ) z ...z dξ ...dξ
1
1
L1
1
r
r
1
r
1





r
r
1
r
Lr
(1.8)
where
mi
ni
(i)
(i)
(i )
(i)
∏ Γ(d j − δ j ξi ) ∏ Γ(1 − c j + γ j ξi )
j =1
j =1
φi (ξi ) =
qi
pi
(i )
(i)
(i)
(i)
Γ
(1
−
d
+
)
δ
ξ
∏
∏ Γ(c j − γ j ξi )
j
j
i
j = mi + 1
j = ni + 1
(1.9)
r
n
(i)
∏ Γ(1 − a j + ∑ α j ξi )
i =1
j =1
, ∀i ∈ {1,..., r}
ψ (ξ1 ,..., ξ r ) =
r
r
q
p
(i )
(i)
∏ Γ(1 − b j + ∑ β j ξi ) ∏ Γ(a j − ∑ α j ξi )
i =1
i =1
j =1
j = n +1
(1.10)
It is assumed that the various H-functions of several variables occurring in the
paper always satisfy the appropriate existence and convergence conditions corresponding
appropriately to those recorded in the book by Srivastava et.al [9,pp.251-253,Eqn’s (C.4)(C.6)]. In case r=2, it reduce to the H-function of two variables [9, p.82, Eqn. (6.1.1)].
P. AGARWAL, S. JAIN
2.
MAIN INTEGRAL FORMULAE
FIRST INTEGRAL

1 1  1 − x α1  1 − y  β1 1-xy
(α , β )  1 − x
y
R
∫ ∫
n
 


 1 − xy
0 0  1 − xy   1 − xy  (1-x)(1-y)

S
m1
n1

 1 − x
 1 − xy

ξ′

y

1
 1− y 


 1 − xy 
=









λ′ 
1



ξ
 1  1− y 
y 

  1 − xy 
λ1

; a, b, c, d ; p, q; γ , δ ;1


µ1
ν1 
 
 z1  1 − x y   1 − y  
  1 − xy   1 − xy  
0, N1 : 1,N 2 ;1, N 3


H

 dxdy
P1 ,Q1 : P2 , Q 2 ; P3 , Q3 
µ2
ν2 
z  1− x y   1− y  
 2  1 − xy   1 − xy  
 
 
 
[n1 / m1 ] ( −n1 )m k
θ1∗ ( v, u , t , e )
1
A
∑
∑
n1 , k Γ(ν − δ n)
k!
v, u , t , e k = 0
0,N1 + 2
: 1, N 2 ; 1, N 3 ;1,1
H
P1 + 2, Q1 + 1:P2 , Q 2 ; P3 , Q3 ;1,1


z1 T : (c , γ )
;(
e
,
E
)
;
1
−
ν
+
δ
n
,1
(
) 
1
j
j
j
j
1, P2
1, P3


z2
T2 : (d j , δ j )
;( f j , Fj )
; ( 0,1)

1, Q 2
1, Q3
c

d

(2.1)
Where
θ
∗
1
is given by (1.4) and
(i)
T1 ≡ (1 − α1 − ξ1R ′-ξ ′k; µ1 , µ2 , ξ1q ), (1 − β1 − λ1 R′ − λ ′k ;ν 1 ,ν 2 , λ1q ), (a j ; α j , A j , 0)
1
1
1,P1
T2 ≡ (b j ; β j , B j , 0)1,Q1 ,(1 − α1 − β1 − (ξ1 + λ1 ) R′ − (ξ1′ + λ1′) k ; µ1 +ν 1 , µ2 +ν 2 , ξ1q + λ1q)
(2.2)
(ii) α 1 , β1 , µ1 , µ 2 ,ν 1 , v2 are all positive, and
Re(α1 ) + µ1[Re (d j / δ j )] + µ 2 [Re (f j / F j )] > 0,
Re(β1 ) + ν 1[Re (d j / δ j )] + ν 2 [Re (f j / Fj )] > 0.
(2.3)
NEW FINITE DOUBLE INTEGRAL FORMULAE INVOLVING POLYNOMIALS
AND FUNCTIONS OF GENERAL NATURE
(iii) Min
{ξ , λ , ξ ′, λ ′, µ ,ν }≥ 0 , (not all zero simultaneously)
1
1
1
1
1
1
(2.4)
(iv)
c
<1
d
(2.5)
SECOND INTEGRAL
ξ
λ


1 1  1 − x α1  1 − y  β1 1-xy
1− x  1  1− y  1
R (nα ,β ) 
y 
y 
∫ ∫

 ; a, b, c, d ; p, q; γ , δ ;1
 1 − xy   1 − xy 

0 0  1 − xy   1 − xy  (1-x)(1-y)


S
m1
n1

 1 − x
 1 − xy

ξ′
λ′
 1  1 − y  1 
y 

  1 − xy  
=
H
 
 z1  1 − x
P2 , Q 2   1 − xy
1,N 2

y

µ1
ν1 
 1− y 


 1 − xy 
 dxdy


[n1 / m1 ] ( −n1 )m k
θ1∗ ( v, u , t , e )
1
A
∑
∑
n1 , k Γ(ν − δ n)
k!
v, u , t , e k = 0

∗
0, 2 : 1, N 2 ;1,1  z1 T1 : (c j , γ j )1, P2 ; (1 −ν + δ n,1)

H
c
2,1:P2 , Q 2 ;1,1  d
T2∗ : (d j , δ j )
; ( 0,1)

1, Q 2






(2.6)
Where
T1∗ ≡ (1 − α1 − ξ1R ′-ξ ′k; µ1 , ξ1q ), (1 − β1 − λ1 R′ − λ ′k ;ν 1 , λ1q )
1
1
T2∗ ≡ (1 − α1 − β1 − (ξ1 + λ1 ) R′ − (ξ1′ + λ1′) k ; µ1 +ν 1 , ξ1q + λ1q)
(2.7)
The conditions of validity of (2.6) easily follow from those given in (2.1).
Proof: To evaluate the integral formula (2.1) we express R(nα , β ) [x] and
S [ x]
m
n
in its
series form with help of (1.3) and (1.7) respectively, change the order of integration and
summation and put the value of H [ z1 , z2 ] in terms of Mellin-Barnes contour integral by
the application of (1.8).Next, we express the binomial terms obtained in the process in
their contour integral form [9, p.18, Eq.(2.6.4)] and change the order of integration, we
have :
[n1 / m1 ] (− n1 )m k
θ1∗ ( v, u , t , e )
1
= ∑
A
∑
n1 , k Γ(ν − δ n)
k!
v, u, t , e k = 0
α +ξ R ′+ξ ′k+ µ ξ + µ η +ξ qs
 1 3
Γ (ν − δ n + s )  1 1  1 − x  1 1 1 1 2 1

y
 
 ∫ ∫ ∫ φ (ξ ,η ) θ1 (ξ ) θ 2 (η )
Γ ( s + 1)  ∫0 ∫0  1 − xy 
 2π i  L1 L2 L3

P. AGARWAL, S. JAIN
 1− y 


 1 − xy 
β1 + λ1 R ′ + λ1′k +ν 1ξ +ν 2η + λ1qs )

 ξ η  c  s
1 − xy
dxdy  z1 z2   d ξ dη ds 
(1 − x )(1 − y )
d


(2.8)
Further, integrate the double integral in (2.8) with the help of the following result [2,
p.415]:
11  1 − x
∫ ∫
0 0  1 − xy
α
β
  1− y 
1-xy
Γ (α ) Γ ( β )
y 
dx dy =

Γ (α + β )
  1 − xy  (1-x)(1-y)
(2.9)
where Re (α) > 0, Re (β) > 0.
Finally, interpreting theξ, η & s contour integral thus obtains in terms of the Hfunction of three variables [9, p.251, Eqn. (C.1)], we arrive at the right hand side of (2.1).
The derivation of the formula (2.6) is similar to (2.1).
3.
SPECIAL CASES OF MAIN INTEGRALS
On account of the most general nature of H-function of two and one variables,
m
(α , β )
[ x] occurring in our formulae given by (2.1) and (2.6), a large number
Rn
(i)
( x )andS
n
of integrals involving simpler functions of one and two variables can be easily obtained as
their special cases. We however gave here only eight special cases by way of illustration:
By applying our results given in (2.1) and (2.6) to the case of Hermite polynomial [10] and
[12 ] and by setting
 1 
H n1 
,
2 x 
= (−1) k ,we have the following interesting consequences of the
S [ x] → x
2
n1 /2
n1
in which case m1 = 2, An ,k
1
main results.
(i)
1 1  1 − x α 1  1 − y
y
∫ ∫

 1 − xy
0 0  1 − xy 
β
 1
1 -x y

(1 -x )(1 -y)

ξ1′ n1
λ1′ n1
ξ
λ




1− x  1  1− y  1
1
−
1
−
x
y
2
y 
y 2 
Rn

 ; a , b , c , d ; p , q ; γ , δ ;1  

  1 − xy   1 − xy 
  1 − xy 
1 − xy 



µ1
ν



 1 − y  1 
 z  1 − x y 

 1, N :1 ,N ; 1 ,N


 1  1 − xy 

1
2
3


1
 1 − xy 
Hn 

 d xd y
H
′
1
λ
µ
ν
  1 − x  ξ 1′  1 − y  1  P1 ,Q 1 : P2 ,Q 2 ; P3 ,Q 3 
 1− x  2  1− y  2 
y
y
 z2 

2 



 
 1 − xy 
 1 − xy 
  1 − xy   1 − xy  


(α , β )  
[n1 / m1 ] ( −n1 ) ( −1) θ ∗ ( v, u , t , e )
1
2k
= ∑
∑
k!
Γ(ν − δ n)
v, u , t , e k = 0
k
NEW FINITE DOUBLE INTEGRAL FORMULAE INVOLVING POLYNOMIALS
AND FUNCTIONS OF GENERAL NATURE


0, N1 + 2 :1, N 2 ;1, N3 ;1,1

H

P1 + 2, Q1 + 1:P2 , Q 2 ;P3 ,Q3 ;1,1 


z1
z2
c
d

T1 : (c j , γ j )
; ( e j , E j ) ;(1 −ν + δ n,1) 
1, P3
1, P2



T2 : (d j , δ j )
; ( f j , Fj ) ; (0,1)

1,
Q
1, Q 2
3

(3.1)
The conditions of validity of (3.1) easily follow from those given in (2.1).
(ii)
ξ
λ


1 1  1 − x α1  1 − y  β1 1-xy
 1  1− y  1
(α , β )  1 − x

γ
δ
R
;
,
,
,
;
,
;
,
;1
y
y
a
b
c
d
p
q
∫ ∫
n
 


 

 1 − xy   1 − xy 

0 0  1 − xy   1 − xy  (1-x)(1-y)


ξ1′ n1
λ1′ n1
 1− x  2  1− y  2
y



 1 − xy 
 1 − xy 
H
Hn
1



1

λ′
  1 − x ξ1′  1 − y  1
2
y
  1 − xy   1 − xy 
 

 







µ1
ν1 
 
 z1  1 − x y   1 − y   dxdy
P2 , Q 2   1 − xy   1 − xy  
1,N 2
[n1 / m1 ] (−n1 ) ( −1)
2k
= ∑
∑
k!
v, u , t , e k = 0
k
θ1∗ ( v, u , t , e )
Γ(ν − δ n)

∗
0, 2 : 1, N 2 ;1,1  z1 T1 : (c j , γ j )1, P2 ; (1 −ν + δ n,1)

H
c
2,1:P2 , Q 2 ;1,1  d
T2∗ : (d j , δ j )
; ( 0,1)

1, Q 2






(3.2)
The conditions of validity of (3.2) easily follow from those given in (2.6).
(2)
For the Laguerre polynomials ([10] and [12]) setting S n′1 ( x ) → Ln1
(α ′)
in which
1
case m = 1, A =  n1 + α ′ 
and the results (2.1) and (2.6) reduce to the following

 ′
1
n1 , k
n
+
1
α
(
)
 1

k
formulae:
P. AGARWAL, S. JAIN
(iii)
ξ
λ


1 1  1 − x α1  1 − y  β1 1-xy
 1− x  1  1− y  1
α ,β
y 
y 
R (n ) 
∫ ∫

 ; a, b, c, d ; p, q; γ , δ ;1
 1 − xy   1 − xy 

0 0  1 − xy   1 − xy  (1-x)(1-y)


ξ′
λ′ 

α′  1 − x  1  1 − y  1
L n11  1 − xy y   1 − xy  


=
0, N1 : 1,N 2 ;1, N 3
H
P1 ,Q1 : P2 , Q 2 ; P3 , Q 3
 1 − x µ1 1 − y ν 1
 

 z1 
y 

−
−
1
xy
1
xy
 
 

µ
ν

1− x  2  1− y  2
 z 2 
y


  1 − xy   1 − xy 


 dxdy



[n1 ] (− n1 )  n + α ′ 
θ1∗ ( v, u , t , e )
1
1
k
∑
∑


k!  n1
 (α ′ + 1)k Γ(ν − δ n)
v, u , t , e k = 0



0,N1 + 2
: 1, N 2 ; 1, N3 ;1,1 
H

P1 + 2, Q1 + 1:P2 , Q 2 ; P3 , Q3 ;1,1 





T1 : (c j , γ j )
;(e j , E j )
; (1 −ν + δ n,1) 
1, P2
1, P3

z1


z2
T2 : (d j , δ j )
;( f j , F j )
; ( 0,1)

1, Q 2
1, Q3
c

d

(3.3)
The conditions of validity of (3.3) easily follow from those given in (2.1).
(iv)
ξ1
λ1


1 1  1 − x α1  1 − y  β1 1-xy
(α , β )   1 − x   1 − y 

y 
R
y
;
a
,
b
,
c
,
d
;
p
,
q
;
,
;1
γ
δ
∫ ∫

n

 



1
−
xy
1
−
xy
(1-x)(1-y)
1
−
xy
1
−
xy








00


ξ′
λ′ 

 1 − x y  1  1 − y  1 
L n1  1 − xy   1 − xy  


α′
=
H
µ1
ν1 

 z1  1 − x y   1 − y   dxdy
P2 , Q2   1 − xy   1 − xy  
1,N 2
[n1 ] (−n1 )  n + α ′ 
θ1∗ ( v, u , t , e )
1
1
k
∑
∑


k!  n1
 (α ′ + 1)k Γ (ν − δ n)
v, u , t , e k = 0

∗
0, 2 : 1, N 2 ;1,1  z1 T1 : (c j , γ j )1, P2 ; (1 −ν + δ n,1)

H
c
2,1:P2 , Q 2 ;1,1  d
T2∗ : (d j , δ j )
; ( 0,1)

1, Q 2






NEW FINITE DOUBLE INTEGRAL FORMULAE INVOLVING POLYNOMIALS
AND FUNCTIONS OF GENERAL NATURE
(3.4)
The conditions of validity of (3.4) easily follow from those given in (2.6).
(3) For Jacobi polynomials ([4] and [7]) setting Rn(
α ,β )
a = b = d = p = q = γ = δ = 1, c = −1, kn′ =
to the following formulae:
(v)
α
β1
11
 1− x
∫ ∫
0 0  1 − xy
ξ′
λ′ 

 1  1− y  1
m1  1 − x
S n1  1 − xy y   1 − xy  


β ,α )
[ x ] in which case
n
2 n!
( −1)

1-xy
1− x
β ,α
Pn( ) 
 1 − xy
(1-x)(1-y)

 1  1− y 
y 

  1 − xy 
[ x ] → Pn(
n
ξ1

y

and the results (2.1) and (2.6) reduce
 1− y 


 1 − xy 
λ1 



µ1
ν1
 
 z1  1 − x y   1 − y 
0, N1 : 1,N 2 ;1, N 3
  1 − xy   1 − xy 
H

P1 ,Q1 : P2 , Q 2 ; P3 , Q3   1 − x  µ 2  1 − y ν 2
y 
 z2 

  1 − xy   1 − xy 



 dxdy



[n1 / m1 ] (−n1 )m k
(1 + β )n θ1∗ ( v, t )
1
A
∑
n1 , k n !Γ(ν − δ n)
k!
ν =0 t =0 k = 0
n
∞
= ∑∑



0,N1 + 2
: 1, N 2 ; 1, N 3 ;1,1 
H

P1 + 2, Q1 + 1:P2 , Q 2 ; P3 , Q3 ;1,1 





z1 T : (c , γ )
;(e j , E j )
; (1 −ν + δ n,1) 
j
1
j
1, P2
1, P3



z2
T2 : (d j , δ j )
;( f j , Fj )
; ( 0,1)

1,Q 2
1, Q3
c


d
(3.5)
Where
( −1)t ( −n )ν (ν )t (1 + α + β + n )ν
∗
θ ( v, t ) =
1
t !2ν (1 + β )ν
The conditions of validity of (3.5) easily follow from those given in (2.1).
P. AGARWAL, S. JAIN
(vi)
1 1  1− x
∫ ∫
0 0  1 − xy
S
m1
n1
α
 1  1− y 
y 

  1 − xy 
β1

1-xy
1− x
β ,α
Pn( ) 

(1-x)(1-y)
1 − xy

ξ′
λ′
 1  1 − y  1 
y 

  1 − xy  

 1 − x
 1 − xy

H
 
 z1  1 − x
P2 , Q 2   1 − xy
1,N 2
ξ
 1  1− y 
y 

  1 − xy 

y

µ1
λ1 



ν1 
 1− y 


 1 − xy 
 dxdy


[n1 / m1 ] (− n1 )m k
(1 + β )n θ1∗ ( v, t )
1
A
= ∑∑ ∑
n1 , k n !Γ (ν − δ n)
k!
ν =0 t = 0 k = 0
n
∞

∗
0, 2 : 1, N 2 ;1,1  z1 T1 : (c j , γ j )1, P2 ; (1 −ν + δ n,1)

H
c
2,1:P2 , Q 2 ;1,1  d
T2∗ : (d j , δ j )
; ( 0,1)

1, Q 2






(3.6)
The conditions of validity of (3.6) easily follow from those given in (2.6).
(4) For generalized polynomial set ([5] and [6]) setting
α ,β
α , β ,τ )
Rn( ) [ x] → Sn(
[ x] in which case p = d = 1, c = −τ , kn′ = 1, β → β / τ and the results
(2.1) and (2.6) reduce to the following formulae:
(vii)

1 1  1 − x α1  1 − y  β1 1-xy
(α , β ,τ )   1 − x
y
S
∫ ∫
 

n

 1 − xy
0 0  1 − xy   1 − xy  (1-x)(1-y)

S
m1
n1

 1 − x
 1 − xy

ξ′

y

1
=
 1− y 


 1 − xy 
λ′ 
1



ξ
 1  1− y 
y 

  1 − xy 
λ1 



µ1
ν1 
 
 z1  1 − x y   1 − y  
  1 − xy   1 − xy  
0, N1 : 1,N 2 ;1, N 3


H

 dxdy
P1 ,Q1 : P2 , Q 2 ; P3 , Q3 
µ
ν 
z  1− x y  2  1− y  2 
 2  1 − xy   1 − xy  
 
 
 
[n1 / m1 ] (− n1 )m k
θ1∗∗ ( v, u , t , e )
1
A
∑ ∑
n1 , k Γ(ν − δ n)
k!
ν ,u ,t , e k = 0
NEW FINITE DOUBLE INTEGRAL FORMULAE INVOLVING POLYNOMIALS
AND FUNCTIONS OF GENERAL NATURE



0,N1 + 2
: 1, N 2 ; 1, N 3 ;1,1 
H

P1 + 2, Q1 + 1:P2 , Q 2 ; P3 , Q3 ;1,1 





z1 T : (c , γ )
;(e j , E j )
; (1 −ν + δ n,1) 
1
j
j
1, P2
1, P3



z2
T2 : (d j , δ j )
;( f j , F j )
; ( 0,1)

1,
Q
1,
Q
2
3
c


d
(3.7)
Where
γ n k n −1 t − v −t
( ) ( )u ( )e
b
θ ∗∗ ( v, u , t , e ) =
v !u !t !e !
1
(α )t ( −τ )
ν
a
 
b
t
( −α − γ n )e
e + λ + qu 
 pe + λ + qu  .
( − β / τ − δ n )ν 

 .R′ = kn + qν + t , 
k
k
(1 − α − t )e

n

n
(3.8)
The conditions of validity of (3.7) easily follow from those given in (2.1).
(viii)
1 1  1− x
∫ ∫
0 0  1 − xy
S
m1
n1
α
 1  1− y 
y 

  1 − xy 

 1 − x
 1 − xy

β1

1-xy
(α , β ,τ )  1 − x
Sn

 1 − xy
(1-x)(1-y)

ξ′
λ′
 1  1 − y  1 
y 

  1 − xy  
H
 
 z1  1 − x
P2 , Q 2   1 − xy
1,N 2
ξ
 1  1− y 
y 

  1 − xy 

y

µ1
λ1 



ν1 
 1− y 


 1 − xy 
 dxdy


[n1 / m1 ] (− n1 )m k
θ1∗∗ ( v, u , t , e )
1
= ∑
A
∑
n1 , k Γ(ν − δ n)
k!
ν ,u ,t , e k = 0

∗
0, 2 : 1, N 2 ;1,1  z1 T1 : (c j , γ j )1, P2 ; (1 −ν + δ n,1)

H
c
2,1:P2 , Q 2 ;1,1  d
T2∗ : (d j , δ j )
; ( 0,1)

1, Q 2






(3.9)
P. AGARWAL, S. JAIN
The conditions of validity of (3.9) easily follow from those given in (2.6).
4.
ACKNOWLEDGMENT
The authors are thankful to worthy referee for his valuable suggestation and Prof. H. M.
Srivastava, University of Victoria, Canada for his valuable research papers which provide
correct approach in the preparation of this paper.
NEW FINITE DOUBLE INTEGRAL FORMULAE INVOLVING POLYNOMIALS
AND FUNCTIONS OF GENERAL NATURE
5.
REFERENCES
1.
Agarwal, B.D. and Chaubey, J.P. , Operational derivation of generating relations for generalized
polynomials, Indian J.Pure Appl.Math.,11(1980), 1155-1157.
2.
Edwards, J., A Treatise on Integral Calculus,Chelsea Pub.Co., Chichester, Brisbane and Toronto,2(1922).
3.
Gupta, K.C. and Gupta, T., On unified eulerian type integrals having general arguments, Soochow J.of
Mathematics, 31(4): (2005), 543-548.
4.
Rainville, E.D., Special Functions, Chelsea Publ. Co., Bronx, New York, (1971).
5.
Raizada, S. K., A Study of unified representation of special functions of mathematical physics and their use
6.
Saigo, M., Goyal, S.P. and Saxena, S., A theorem relating a generalized Weyl fractional integral, Laplace
in statistical and boundary value problems, Ph.D. thesis, Bundelkhand Univ., Jhansi, India,(1991).
and Varma transforms with applications, J. Fractional Calculus,13(1988),43-56.
7.
Salim, T.O., A series formula of a generalized class of polynomials associated with Laplace transform and
fractional integral operators, J.Rajasthan Acad. Phy. Sci.,1(3) : (2002),167-176.
8.
Srivastava, H.M., A contour integral involving Fox’s H- function, Indian J.Math.14 (1972), 1-6.
9.
Srivastava, H.M., Gupta, K.C. and Goyal, S.P., The H-Function of One and Two Variables with
Applications, South Asian Publishers, New Delhi, Madras, (1982).
10.
11.
12.
Srivastava, H.M. and Singh, N.P.,The integration n of certain products of the multivariable H-function
with a general class of polynomials,Rend. Circ. Mat. Palermo, 32 (1983), 157-187.
Srivastava, H. M. and Manocha, H.L., A Treatise on Generating Functions, Halsted Press (Ellis Horward
Limited, Chichester), John Wiley and Sons, New York, Chichester,Brisbane and Toronto,(1984).
Szego, C., Orthogonal Polynomials.Amer.Math.Soc.Colloq.Publ.vol.23, 4th Ed., Amer.Math.Soc,
Providence, Rhode Island (1975).
Praveen Agrawal
Department of Mathematics,
Anand International College of Engineering,
Jaipur-303012,
India
Shilpi Jain
Deparment of Mathematics,
Poornima College of Engineering,
Jaipur-302017,
India
Email: [email protected]
Received August 2010
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