Generalized Fractional Integral Operators Involving Mittag-Leffler Function

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.

Fractional Integral and Derivative Formulas by Using Marichev-Saigo-Maeda Operators Involving the S-Function

“…Further, the S-function defined in (43) possesses the lead that a number of -Mittag-Leffler functions, K-function, M-series, and Mittag-Leffler function happen to be the particular cases of this function. Some special cases of fractional calculus involved as above said function have been explored in the literature by a numeral of authors ( [35][36][37][38][39][40][41]) with different arguments. Therefore, results presented in this paper are easily converted in terms of a comparable type of novel interesting integrals with diverse arguments after various suitable parametric replacements.…”

Generalized Fractional Integral Formulas for the k-Bessel Function

“…Further, the generalized Bessel function defined in (1) possesses the lead that a number of Bessel functions, trigonometric functions, and hyperbolic functions happen to be the particular cases of this function. Some special cases of integrals involving generalized Bessel function have been explored in the literature by a number of authors ( [20][21][22][23][24][25][26]) with different arguments. Therefore, results presented in this paper are easily converted in terms of a comparable type of novel interesting integrals with diverse arguments after various suitable parametric replacements.…”

Application of Laplace Transform on Fractional Kinetic Equation Pertaining to the Generalized Galué Type Struve Function

“…Particularly, the kinetic equations define the continuity of motion of substance and are the elementary equations of mathematical physics and natural science. The extension and generality of fractional kinetic equations and various fractional operators with special functions were found (Agarwal et al [1], Amsalu and Suthar [2], Baleanu et al [3,4], Chaurasia and Pandey [5], Choi and Agarwal [6,7], Zaslavsky [8], Gupta and Parihar [9], Gupta and Sharma [10], Haubold and Mathai [11], Kumar et al [12], Nisar et al [13], Saxena et al [14][15][16], Saichev and Zaslavsky [17], Suthar et al [18], and Tariboon et al [19]). In view of the effectiveness and a great significance of the kinetic equation in some astrophysical problems the authors develop a further generalized form of the fractional kinetic equation involving generalized Galué type Struve function.…”