Extension of Mittag-Leffler function

TL;DR

This paper introduces an extended version of the Mittag-Leffler function using extended beta functions, providing integral representations, Mellin transforms, and connections to extended fractional derivatives.

Contribution

It presents a novel extension of the Mittag-Leffler function based on extended beta functions, with new integral and Mellin transform representations.

Findings

Derived integral representations of the extended Mittag-Leffler function

Obtained Mellin transform in terms of Wright hypergeometric function

Linked extended fractional derivatives to the new function

Abstract

In this paper, we present an extension of Mittag-Leffler function by using the extension of beta functions (\"{O}zergin et al. in J. Comput. Appl. Math. 235 (2011), 4601-4610) and obtain some integral representation of this newly defined function. Also, we present the Mellin transform of this function in terms of Wright hypergeometric function. Furthermore, we show that the extended fractional derivative of the usual Mittag-Leffler function gives the extension of Mittag-Leffler function.