Certain Type of Special Function Associate by Pathway Fraction Integral Operator

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 Mittag-Leffler function, G-function.

1. INTRODUCTION

   The Pathway fractional integral operator introduced by Nair

   [9] is defined in the following manner

   for < 1it is a finite range density with [1 (1

   )||]1 and (1.2) remains in the extended generalized type-1 beta family.

   () = ||1[1 + ( 1)||]1 … (1.6)

   Provided that < < , > 0, 0, > 1which is the extended generalized type-2 beta model for real x. It includes thetype-2 beta density, the F-density, 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)

   Riemann-Liouville fractional integral operator.

   1

   = 1 () , for 1 … (1.5)

   The following generalized M-series was introduced by Sharma and Jain [10]:

   2 ()

   ,

   1

   1

   ( ,

   ; , ; ) = ()

   1

   The pathway density in (1.2) for <1, includes the extended type-1 beta density, the triangular density, the uniform density and many other p.d.f. for >1, we have

   Theorem: 1

2. 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 H-function[4] was studied by Skibiski [13] and defined

(,) [1 ,

as a following manner:

,[] = , [ |(, )] =

0+

(1) () (1)

,

( , )

,

,

,

(, )

=1

=0

!

+ (1 + )

,

( , )

= () 1

… (1.10)

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 Mittag-Leffler 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 G-functions 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 theorem-1 then we get the known result of

2. If we take , 0, and k=1 in theorem-1 then we get the known result of [9, eq. (25) p.244].

3. If we take 0, 0, = = 1 in theorem-1 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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