Generalized Fractional Integral Operators Involving Mittag-Leffler Function

Hafte Amsalu,D L Suthar

doi:10.1155/2018/7034124

Abstract

The aim of this paper is to study various properties of Mittag-Leffler (M-L) function. Here we establish two theorems which give the image of this M-L function under the generalized fractional integral operators involving Fox’s H-function as kernel. Corresponding assertions in terms of Euler, Mellin, Laplace, Whittaker, and K-transforms are also presented. On account of general nature of M-L function a number of results involving special functions can be obtained merely by giving particular values for the parameters.

Highlights

We recall the definition of generalized fractional integral operators involving Fox’s H-function as kernel, defined by Saxena and Kumbhat [4] means of the following equations: Rxμ,rα
We have investigated and studied two classes of generalized fractional integral operators involving Fox’s H-function as kernel due to Saxena and Kumbhat which are applied on M-L function
We discussed the actions of fractional integral operators under Euler, Mellin, Laplace, Whittaker, and K-transforms and results are given in better pragmatic series solutions

Summary

Introduction and Preliminaries

We recall the definition of generalized fractional integral operators involving Fox’s H-function as kernel, defined by Saxena and Kumbhat [4] means of the following equations: Rxμ,,rα [f (x)]. We consider two generalized fractional integral operators involving the Fox’s H-function as the kernels and derived the following theorems. Changing the order of the integration valid under the condition given with the theorem, we obtain l rx−μ−rα−1 ∑ n=0 Γ (γ)n an (λn + β). Let λ, β, θ, γ ∈ C, x > 0, R(λ) > 0, R(θ) < 1, f(x) ∈ LP(0, ∞), 1 ≤ p ≤ 2, | arg k| < λπ/2, λ > 0, and a ∈ C; the fractional integration Kxε,,αr of the product of M-L function exists, under the condition p−1 + q−1 = 1,. Changing the order of the integration valid under the condition given with the theorem statement, we obtain

Integral Transforms of Fractional Integral Involving M-L Function

Properties of Integral Operators

Conclusions