Extended (p,q)-Mittag-Leffler function and its properties

TL;DR

This paper introduces an extended (p,q)-Mittag-Leffler function via beta function extensions, explores its integral and Mellin transform representations, and links it to extended fractional derivatives, broadening the classical ML function's scope.

Contribution

It defines a new extended (p,q)-Mittag-Leffler function and derives its integral and Mellin transform representations, connecting it to extended fractional derivatives.

Findings

Derived the integral representation of the extended (p,q)-ML function.

Obtained the Mellin transform in terms of Wright hypergeometric function.

Connected the extended fractional derivative to the new ML function.

Abstract

In this study our aim to define the extended $(p,q)$-Mittag-Leffler(ML) function by using extension of beta functions and to obtain the integral representation of new function. We also take the Mellin transform of this new function in terms of Wright hypergeometric function. Extended fractional derivative of the classical Mittag-Leffler(ML) function leads the extended (p,q)-Mittag-Leffler(ML) function.