Generalized Fractional Integral Operators andM-Series

A M Khan,Amit Chouhan,R K Kumbhat,Anita Alaria

doi:10.1155/2016/2872185

Abstract

Two fractional integral operators associated with FoxH-function due to Saxena and Kumbhat are applied toM-series, which is an extension of both Mittag-Leffler function and generalized hypergeometric functionpFq. The Mellin and Whittaker transforms are obtained for these compositional operators withM-series. Further some interesting properties have been established including power function and Riemann-Liouville fractional integral operators. The results are expressed in terms ofH-function, which are in compact form suitable for numerical computation. Special cases of the results are also pointed out in the form of lemmas and corollaries.

Highlights

Introduction and PreliminariesThe subject of fractional calculus, which deals with investigations of integrals and derivatives, has gained importance and popularity during the last four decades
Different extensions of various fractional integrations operators are studied by Kalla [1, 2], McBride [3], Kilbas [4, 5], Kiryakova [6], Purohit and Kalla [7], Kumbhat and Khan [8], and so forth
Theorem 8 readily follows on evaluating the Mellin transform of t−ρ−δ−Vi by means of the formula given by Erdelyi [18]

Summary

Introduction and Preliminaries

The subject of fractional calculus, which deals with investigations of integrals and derivatives, has gained importance and popularity during the last four decades. Saxena and Kumbhat [9] defined the fractional integration operators by means of the following equations: Rxη,,αr [f (x)]. The conditions of the validity of these operators are as follows:. In (1) and (2) HPM,Q,N(x) denotes H-function introduced and defined by Fox [10] via a Mellin-Barnes type integral as HPM,Q,N (z) ≡ HPM,Q,N [ z [. Series (5) is defined when none of the parameters bj’s, j = 1, 2, . When p = q + 1 and |z| = δ, the series can converge on conditions depending on the parameters. Properties of M-series are further studied by Saxena [13], Chouhan and Saraswat [14], and so forth. The right sided Riemann-Liouville fractional integral operator Iaλ+ is defined by Samko et al [16] for Re(λ) > 0, as (Iaλ+f) (x). Γ (1 − λ + σ) where Re(μ±σ) > −1/2 and Wλ,μ(⋅) is the Whittaker function [17, 18] defined as

Operators and M-Series

Mellin and Whittaker Transforms

Properties of Integral Operators

Conclusions