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 Authors : Hemlata Saxena, Himanshu Sharma, Harsha Babani
 Paper ID : IJERTCONV3IS31002
 Volume & Issue : ATCSMT – 2015 (Volume 3 – Issue 31)
 Published (First Online): 24042018
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Certain Type of Special Function Associate by Pathway Fraction Integral Operator
Hemlata Saxena1, Himanshu Sharma2, Harsha Babani3
Department of Mathematics, Career Point University, Kota
Abstract: In the present paper we consider product of some special function associated with the pathway functional integral operator. This operator generalizes of the classical Riemann Liouville fractional integral operator. The results derived here are quite general in nature and their several known and new special cases are also obtained here.
Keywords: Pathway fractional integral operator, Fox's H function, Generalized MittagLeffler function, Gfunction.
INTRODUCTION
The Pathway fractional integral operator introduced by Nair
[9] is defined in the following mannerfor < 1it is a finite range density with [1 (1
)]1 and (1.2) remains in the extended generalized type1 beta family.
() = 1[1 + ( 1)]1 … (1.6)
Provided that < < , > 0, 0, > 1which is the extended generalized type2 beta model for real x. It includes thetype2 beta density, the Fdensity, the Cauchy density and many more.
Here we consider only the case of pathway parameter < 1. For 1,(1.2) and (1.6)take the exponential from, since
(,)
[ ](1)
(0+
) () = (1) [1
]1 () … (1.1)
1 [1 (1 )]1
1
0
0
 1
( )

wheref(x) L(a, b), C, R() > 0, > 0 and pathway parameter <1.
=
1
[1 + 1] 1
The pathway model introduced byMathai [6] and further studied bythe Mathai and Haubold[7], [8]. For real scalar, the pathway model for scalar random variables is represented by the following probability density function (p. d. f.).
() = 1[1 (1 )]1… (1.2)
= 1 … (1.7)
This include the generalized Gamma, the Weibull, the Chi square, the Laplace, Maxwell
Boltzmann and other related densities.
When 1, [1
(1)]1
U, the operator (1.1)
< < , > 0, 0, [1 (1 )x]1 > 0, >
reduces to theLaplace integral transform of f with parametera
0whereCis the normalizing constant and is called the pathway parameter. For real , thenormalizing constant is as follows:
:
0+
0+
((,)
) () =
() =
x
()
[(1) (+ +1)]
0
0
… (1.8)
1
1
=
2
() ( +1) , < 1 .. (1.3)
when = 0, a = 1then replacing by 1 in (1.1) the integral
1
operator reduces to the
1
1
1
1
[(1) ( )]
= 2 () ( )
, for 1
1
> 0, > 1 … (1.4)
RiemannLiouville fractional integral operator.
1
= 1 () , for 1 … (1.5)
The following generalized Mseries was introduced by Sharma and Jain [10]:
2 ()
,
1
1
( ,
; , ; ) = ()
1
The pathway density in (1.2) for <1, includes the extended type1 beta density, the triangular density, the uniform density and many other p.d.f. for >1, we have
Theorem: 1
MAIN RESULTS
( ).( )
Let , , , , 1, 2 > 0, () > 0, () > 0, () >
,
() = 0 1
0, (1 +
) > , [0, ()], , , , <
=
(1)…() (+)
… (1.9)
1
1, [ + ] > 0, [ + ]   <
= ()
1 ,   < 1 ,  < , > 0, =
1 2 1 2 2
where, , , () > 0 , ,  < for other details see [10]
1, , , = 1, ,
Then
Fox Hfunction[4] was studied by Skibiski [13] and defined
(,) [1 ,
[] , [ (, )] ,()] ()as a following manner:
,[] = , [ (, )] =
0+
(1) () (1)
,
( , )
,
,
,
(, )
=1
=0
!
+ (1 + )
,
( , )
= () 1
[  ]… (1.10)
1 [(1 )]
,
[(1 )] (, )where = 1
and (h=1, 2,..N)
(, )
. [
. [
2 2 (1 ) 
(, ) (1 + + , )]
1
and
() =
{ 1 ( + )}{ 1, (1 )}
= =
= =
{ (1 )} { ( + )}
… (2.1)
Proof: Making use (1.9), (1.10), (1.11) and (1.1)in the theorem 1 then interchanging the order of integration and
=+1
= 2()
=+1
summation by means of beta function we at once arrive at the desired result (2.1).
for convergence condition and other details see[4],[11].
A generalized MittagLeffler function studied by Shukla & Prajapati [12] in the following manner
Theorem: 2
Let , , () > 0, () > 0, (1 + ) >
1
0, () > 0, < 1, ,
,() =
()
() , , , () >
1
,
=0 (+)
!
[ + ] > 0, [ + ] > 0   < 2 1,
0, () > 0 … (1.11)
where , , , () > 0, () > 0, () > 0 (0,1) . This is a generalization of the exponential function exp(z), the confluent hyper geometric function
,
,
(, , ). The Mittag Leffler function E (z), the wimenenss function E,(z) and the function E (z)defined by
> 0, 1, 2
Then
2
2
,   < 1 ,
2
> 0, ,  < , = 1, , ,
= 1, ,
Prabhakar.
( + )
(,) [1 ,
[] , [ (, )] []] ()() =
()
,

(, )
,,
0+
0+
Denotes the generalized Pochhamamer symbol which in
+1
dkx+k+qr (1 + ) N
cx
(e , E ) x
particular reduces to
=1 [
] if .
= (k) 1 HM, [  P
P ] G [ ]
The Gfunctions is defined by Lorenzo & Hartley [5]:
1 [a(1 )]k+qr
… (2.2)
P, Q [a(1 )] (fQ, FQ)
q, 1 1, a(1 )
[, ] = 1
()()
Proof:Making use (1.9), (1.10), (1.12) and (1.1)in the
,,
… (1.12)
=0 [(1+)][(+)]
theorem 1 then interchanging the order of integration and summation by means of beta function we at once arrive at the desired result (2.2).
Special Cases:
1. If we take k=1, in theorem1 then we get the known result of
[3, eq. (2.2)].2. If we take , 0, and k=1 in theorem1 then we get the known result of [9, eq. (25) p.244].
3. If we take 0, 0, = = 1 in theorem1 then we get the known result of [9, eq. (26) p.245].
4. If we take 0, 0, = 1 e and in theorem 2 then we get the known result of [10, eq. (3.3)].
CONCLUSION
The pathway fractional integral operator is expected to have wide application in statistical distribution theory. It could help to extend some classical statistical distribution to wider classes of distribution. During the last three decades fraction calculus has been applied to almost every field of science, engineering and mathematics.
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