ON SOME FRACTIONAL INTEGRAL FORMULAE INVOLVING THE
MULTIVARIABLE I -FUNCTION AND A GENERAL CLASS OF POLYNOMIALS
Yashwant Singh1, Harmendra Kumar Mandia2
1,2 Department of Mathematics,
1Governmant college Kaladera,Jaipur(Rajasthan), India
2Seth Motilal (P.G.) College, Jhunjhunu(Rajasthan), India
Abstract— In the present paper, the authors have obtained two fractioal integral formulae involving the product of a general class of polynomials and the multivariable
I
-function. On account of the most general nature of these functions, a number of results (known and new) follow as special cases of our formulae. In the end, we obtain a fractional integral formula involving the laguerre and the hermite polynomials as a simple special case of our main formula.Keywords- Riemann-Liouville Fractional Integral Operator, Multivariable
I
-function, General Class of Polynomials, Mellin- Barnes contour Integral(2000 Mathematics subject classification: 33C99).
I. INTRODUCTION
In the present paper the fractional integral operator will be defined and represented as follows:
( ) 1 ( )
1( ) , Re( ) 0
( )
x
v v
c x
c
I f x x t f t dt v
v
(1.1)
The special case of the above fractional integral for
0
c
is the Riemann-Liouville fractional integral operator and will be written asI
xv f x ( )
.
Also, the fractional integral operator investigated by Erdelyi-Kober will be defined and represented as follows:
1, 1 1
0
( ) ( ) ( ) , Re( ) 0, 0
( )
v x
v v
x
I f x x x t t f t dt v
v
(1.2)
Clearly, this is a generalization of Riemann-Liouville fractional integral operator.
The multivariable
I
-function introduced by Prasad [2] will be define and represent it in the following manner :
2
( )( )( )( )
2 2
0, :...:0, :( ', '):...: , 1
,...,
, :...: , :( ', '):...: ,r r
r
r r
r r
n n m n m n
r p q p q p q p q
I z z I
( ) ( ) ( )
' '' ' ' '
2 2 2 1,2 1, 1, ' 1,( )
( ) ( ) ( )
' '' ' ' '
2 2 2 1,2 1, 1, ' 1,( )
( , , ) :...:( , ,..., ) :( , ) :...:( , )
1
,...,
( , , ) :...:( , ,..., ) :( , ) :...:( , )r r r
j j j p rj rj rj pr j j p j j pr
r r r
j j j q rj rj rj qr j j q j j qr
a a a
r b b b b
z z
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
w
(1.3) Where
( 1) w
( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( ) ( )
1 1
( ) ( ) ( ) ( )
1 1
1
( ) (1, 2,..., )
1
i i
i i
i i
m n
i i i i
j j i j j i
j j
i i q p
i i i i
j j i j j i
j m j n
b s a s
s i r
b s a s
(1.4)
3 2
3 2
2 3
2 3
( ) ( )
2 2 3 3
1 1
1 1
1 2 3
( ) ( )
2 2 3 3
1 1
1 1
1 1
( ,..., )
n n
i i
j j i j j i
i i
j j
r p p
i i
j j i j j i
i i
j n j n
a s a s
s s
a s a s
2
( ) 1 1
2
( ) ( ) ( )
2 2
1 1 1
1 1 1
... 1
... 1 ... 1
r
r r
r
n r
i
rj rj i
j i
p r q q r
i i i
rj rj i j j i rj rj i
i i i
j n j j
a s
a s b s b s
(1.5)
( ) ( ) ( ) ( )
, , , ( 1,..., )( 1,..., )
i i i i
j j kj kj
i r k r
are positive numbers,
( ) ( )
, ( 1,..., ), , ( 2,..., )
i i
j j kj kj
a b i r a b k r
are complex numbers and here
( ) ( ) ( ) ( )
, , , ( 1,..., ), , , ( 2,..., )
i i i i
k k k
m n p q i r n p q k r
are non-negative integers where
( ) ( ) ( ) ( )
0 m
i q
i, 0 n
i p
i, q
k 0, 0 n
k p
k. Here
( ) i
denotes the numbers of dashes. The contoursL
iin the complex
s
i-plane is of the Mellin-Barnes type which runs from
w
to w
with indentations, if necessary, to ensure that all the poles of b
( )ji
( )jis
i j 1,..., m
( )i
are
separated from those of
( ) 1
1 1,...,
r i
rj rj i r
i
a s j n
.
For further details and asymptotic expansion of the
I
-function one can refer by Prasad [2].
In what follows, the multivariable
I
-function defined by [2] will be represented in the contracted notation:
( ) ( )
2
( ) ( )
2 2
0, :...:0, : ', ' :...: ,
, :...: , : ', ' :...: , 1
,...,
r r
r
r r
r r
n n m n m n
p q p q p q p q r
I z z
Or simply by
I z
1,... z
r
.
According to the asymptotic expansion of the gamma function, the counter integral (1.3) is absolutely convergent provided that
arg 1 , 0
i
2
i iz U U
;
i 1 , 2, . . . . , r
(1.6)Where
( ) ( )
( ) ( )
( ) ( ) ( ) ( )
1 1 1 1
i i
i i
i i
p q
n m
i i i i
i j j j j
j j n j j m
U
3 3
2 2
2 3
( ) ( ) ( ) ( )
2 2 3 3
1 1 1 1
( ) ( )
n p
n p
i i i i
j j j j
j j n j j n
( ) ( )
1 1
... ( )
r r
r
n p
i i
rj rj
j j n
3 2
( ) ( ) ( )
2 3
1 1 1
( ... )
q r
q q
i i i
j j rj
j j j
(1.7)
The asymptotic expansion of the
I
-function has been discussed by Prasad [2]. His results run as follow:1
1 1 1
[ ,...,
r] 0( ...
r r), max{ ,...,
r} 0
I z z z
z
z z
Where
( ) ( )
min Re(
i/
i)
i
b
j j
,
1 , . . . . ,
( )ij m
;i 1 , . . . . , r
And1
1 1 1
[ ,...,
r] 0( ...
r r), min{ ,...,
r}
I z z z
z
z z
Where
( )
( )
max Re( 1 )
i j
i i
j
a
;
1 , . . . . ,
( )ij n
,
i 1 , . . . . , r
The details of the function can be found in the paper of Prasad [2].
Again we shall use the following notation:
' ( )
1,1 2
2 2
( ) '
1,
; ,..., ,...
0, :...:0, :...
, :...: , :...
; ,..., ,...
r
j j j u
r
r r
r r
j j j v
z
n n u
p q p u q v z
v
I
For the multivariable
I
-function given by (1.3)Srivastava ([4], p.1, eq. (1)) introduced a general class of polynomials
S
nm[ ] x
defined by means of the following equation:, 0
( )
[ ] , 0,1, 2,...
!
nm
m mk k
n n k
k
S x n A x n
k
(1.8)
Where
m
is an arbitrary positive integer and the coefficientsA
n k,( , n k 0)
are arbitrary constants, real or complex. On specializing these coefficints
A
n k,; S
nmyields a number of known polynomials as special cases. These include, among others, the Hermite polynomials, the Jacobi polynomials, the Laguerre polynomials, the Bessel polynomials, the Gould-Hopper polynomials, the Brafman polynomials and many others.
II. FRACTIONAL INTEGRAL FORMULAE
( ) ( )
m u( ) (
v)
wc
I
xx
x
x
S
n ax x x
1 1 1
' ' ' '
'
( ) ( )
1 u( ) (
v) ,...,
wm u v w
S
n bx x x I z x x x
z x
r ur( x ) (
vrx )
wr
=
' '
'
' ' , ', '
, , 0 0 ' 0
( ) ( ') '
n n
m m
k k
mk m k n k n k
s l t k k
x n n A A a b
' '
' ' 1 ' ' 1
( 1) ( )
! '! ! ! ( ) ( ) !
s s uk u k l t s
k k wk w k
x c x
k k l t s s
1 1 1 1 1 2
2 2
1
( ' ' ; ,..., ),
0, :...:0, 3:...
, :...: 3, 3:...
( ' ' ; ,..., ),
r
v w u
r
v w u
r r r r r
r
r
uk u k l t u u
z x
n n
p q p q
z x
s uk u k l t u u
I
1 1
1 1
( ' '; ,..., ),( ' '; ,..., );...
( ' '; ,..., ),( ' '; ,..., );...
r r
r r
vk v k v v wk w k w w
l vk v k v v t wk w k w w
(2.1)
Provided that
(i)
Re( ) v 0; min u v w u v w u v w
i, ,
i i, , , , ', ', ' 0, i 1, 2,..., ; r
max | arg x ,| arg x
and theI
-functionof
r
variables involved in (2.1) satisfy the usual convergence and existance conditions.(ii)
( )
( )
1 ( ) 1
Re( ) min Re 1 0
i
r i
j
i i
i j m j
u d
(iii) The series occuring on the right-hand side of (2.1) is absolutely convergent.
,
( ) ( )
m u( ) (
v)
wx n
I
x
x
x
S ax x x
1 1 1
' ' ' '
'
( ) ( )
1 u( ) (
v) ,...,
wm u v w
S
n bx x x I z x x x
( ) ( )
r r r
u v w
z x
rx x
=
' '
'
' ' , ', '
, 0 0 0 ' 0
( ) ( ') '
n n
m m
k k
mk m k n k n k
l t k k k
x n n A A a b
' '
' ' 1 ' ' 1
( 1)
! '! ! !
s uk u k l t
k k wk w k
x
k k l t
1 1 1 1 1 2
2 2
1
(1 ' ' ; ,..., ),
0, :...:0, 3:...
, :...: 3, 3:...
( 1 ' ' ; ,..., ),
r
v w u
r
v w u
r r r r r
r
r
uk u k l t u u
z x
n n
p q p q
z x
v uk u k l t u u
I
1 1
1 1
( ' '; ,..., ),( ' '; ,..., );...
( ' '; ,..., ),( ' '; ,..., );...
r r
r r
vk v k v v wk w k w w
l vk v k v v t wk w k w w
(2.2) Where
( )
( )
1 ( ) 1
0, Re( ) min Re
i
r i
j
i j m i
i j
u d
The series occuring on the right-hand side of (2.2) is absolutely convergent and the set of conditions (i) specified in the formula (2.1) are satisfied.
Proof of (2.1):
To establish the fractional integral formula (2.1), we first express the general class of polynomials
S
nm[ ] x
and
'
'
[ ]
m
S
nx
occuring on its left-hand side in the series form given by (1.8) and replace the multivariable
I
-function occuring therein by its well known Mellin-Barnes contour integral [5].The left-hand side of (2.1) (say
) takes the following form:
,0
( )
( ) ( )
!
nm
mk
c x n k
k
I x x x n A
k
' '
' ' ', ' ' 0
( ')
( ) ( )
'!
nm
u v w k m k
n k k
ax x x n A
k
1
' ' ' '
1
( ) ( ) 1 ... ( ,..., )
(2 )
r
u v w k
r r
L L
bx x x
i
1 1 1 1
1
( ).... ( )
1 r rz x
1 u( x ) (
vx )
w
( ) ( )
r 1...
r r r
u v w
r r
z x x x
d d
(2.3)
Now interchanging the order of
i-integrals and the fractional integral involved in (2.3), collecting powers of
, ( )
x x
and( x )
in the expression thus obtained, making use of the following binomial expressions for( x )
and( x )
0
( ) ; | | 1
l
l l
x x
x
0
( ) ; | | 1
t
t t
x x
x
And changing the order of
i-integrals and
l t ,
-series,which is easily seen to be permissible under conditions stated, we get after a little simplification the following result:
' '
' '
, ', '
, 0 0 ' 0
( ) ( ')
! '! '
n n
m m
mk m k
n k n k
l t k k
n n
k k A A
' ' ' '
! !
vk v k l wk w k t
l k
1
1
1 ... ( ,..., ) (2 )
r
r r
L L
i
1 1 1
1
( ).... ( )
1 r rz
1 v w ... z
r vr wr r
' ' 1 1 ...
1 1
1 1
' ' ... 1
' ' ... 1
r r r r
uk u k u u l t v
c x
r r
vk v k v v
I x vk v k v v l
1 11 1
1' ' ... 1
' ' ... 1 ...
r r
r r r
wk w k w w
d d
wk w k w w t
(2.4)
Now making use of the familiar formula ([],p.22)
0
( 1) 1 ( )
, Re( ) 1
( ) 1 !
s s v s
v c x
s
x c x I x
v s s v s
(2.5)
In eq. (2.5), changing the order of
s
-series and
i- integrals, which is also permissible under conditions stated and interpreting the resulting multiple Mellin-Barnes contour integral so obtained in terms of
I
-function ofr
variables, we finally arrive at the desired result.In order to establish the fractional integral formula (2.2) (involving Erdelyi-Kober operator defind by (1.2), we proceed along the lines followed in (2.1) and make use of the result
,v
, Re( )
x
I x x
v
(2.6) In place of formula (2.5).
3. Special Cases
Each of the fractional integral formulae (2.1) and (2.2) has two-fold generality. Not only do the general class of polynomials involved in these results reduce, under special cases, to a large spectrum of polynomials defined by Srivastava [4], the multivariable
I
-function occuring herein can also be suitably specilized to a remarkable wide variety of useful functions. A number of results (known and new) follow as special cases of these formulae.If in (2.1) and (2.2), we put
2
...
r 10
2...
r 1,
2...
r 10
n n
p p
q q
, the multivariable
I
-function reduces to multivariableH
- function and we get result given by Gupta et. al. [1] ( ) ( )
m u( ) (
v)
wc
I
xx
x
x
S
n ax x x
1 1 1
' ' ' '
'
( ) ( )
1 u( ) (
v) ,...,
wm u v w
S
n bx x x H z x x x
z x
r ur( x ) (
vrx )
wr
=
' '
'
' ' , ', '
, , 0 0 ' 0
( ) ( ') '
n n
m m
k k
mk m k n k n k
s l t k k
x n n A A a b
' '
' ' 1 ' ' 1
( 1) ( )
! '! ! ! ( ) ( ) !
s s uk u k l t s
k k wk w k
x c x
k k l t s s
1 1 1 1 1
1
( ' ' ; ,..., ),
3:...
3, 3:...
( ' ' ; ,..., ),
r
v w u
vr wr ur r
r
uk u k l t u u
z x
n
p q
z x
s uk u k l t u u
H
1 1
1 1
( ' '; ,..., ),( ' '; ,..., );...
( ' '; ,..., ),( ' '; ,..., );...
r r
r r
vk v k v v wk w k w w
l vk v k v v t wk w k w w
(3.1)
Provided all the conditions are satisfied given in (2.1), [1].
,
( ) ( )
m u( ) (
v)
wx n
I
x
x
x
S ax x x
1 1 1
' ' ' '
'
( ) ( )
1 u( ) (
v) ,...,
wm u v w
S
n bx x x H z x x x
( ) ( )
r r r
u v w
z x
rx x
=
' '
'
' ' , ', '
, 0 0 0 ' 0
( ) ( ') '
n n
m m
k k
mk m k n k n k
l t k k k
x n n A A a b
' '
' ' 1 ' ' 1
( 1)
! '! ! !
s uk u k l t
k k wk w k
x
k k l t
1 1 1 1 1
1
(1 ' ' ; ,..., ),
3:...
3, 3:...
( 1 ' ' ; ,..., ),
r
v w u
vr wr ur r
r
uk u k l t u u
z x
n
p q
z x
v uk u k l t u u
H
1 1
1 1
( ' '; ,..., ),( ' '; ,..., );...
( ' '; ,..., ),( ' '; ,..., );...
r r
r r
vk v k v v wk w k w w
l vk v k v v t wk w k w w
(3.2)
Provided all the conditions are satisfied given in (2.2), [1].
If we take
c 0
in (3.1) and make use of the following result:
' ( )
1
1,
( ) '
1 1,
; ,..., :
0, :
, 1: ( ; ,..., ), ; ,..., :
0
( 1) ( )( ) !
r
j j j p
r
r j j j
r q
s z a
n
p q z s k u u b
s
s s v s H
= 1
' ( )
1,
' ( )
1 1,
; ,..., :
0, :
, 1: ( ; ,..., ), ; ,..., :
r
j j j
p r
r j j j
r q
z a
n
p q z v k u u b
H
We arrive at the following fractional integral formula after a little simplification:
( ) ( )
m u( ) (
v)
wc
I
xx
x
x
S
n ax x x
1 1 1
' ' ' '
'
( ) ( )
1 u( ) (
v) ,...,
wm u v w
S
n bx x x H z x x x
z x
r ur( x ) (
vrx )
wr
=
' '
'
' ' , ', '
, 0 0 ' 0
( ) ( ') '
n n
m m
k k
mk m k n k n k
l t k k
x n n A A a b
' '
' ' 1 ' ' 1
( 1) ( )
! '! ! !
s s uk u k l t
k k wk w k
x c x
k k l t
1 1 1 1 1
1
( ' ' ; ,..., ),
3:...
3, 3:...
( ' ' ; ,..., ),
r
v w u
vr wr ur r
r
uk u k l t u u
z x
n
p q
z x
v uk u k l t u u
H
1 1
1 1
( ' '; ,..., ),( ' '; ,..., );...
( ' '; ,..., ),( ' '; ,..., );...
r r
r r
vk v k v v wk w k w w
l vk v k v v t wk w k w w
(3.3)
Where the conditions of validity directly obtainable from those mentioned with (2.1) are satisfied.
