On generalized Mittag-Leffler-type functions of two variables

TL;DR

This paper introduces and analyzes generalized Mittag-Leffler functions of two variables, exploring their properties, convergence regions, integral representations, and associated differential equations, expanding the understanding of hypergeometric functions in multiple variables.

Contribution

It develops a new class of two-variable Mittag-Leffler functions, determines their convergence regions, and derives integral representations and differential equations related to these functions.

Findings

Convergence regions of the functions are fully characterized.

Integral representations of Euler type are established.

Laplace transforms and differential equations associated with the functions are derived.

Abstract

We aim to study Mittag-Leffler type functions of two variables ${{D}_{1}}\left( x,y \right),...,{{D}_{5}}\left( x,y \right)$ by analogy with the Appell hypergeometric functions of two variables. Moreover, we targeted functions ${{E}_{1}}\left( x,y \right),$ $...,{{E}_{10}}\left( x,y \right)$ as limiting cases of the functions ${{D}_{1}}\left( x,y \right),$ $...,{{D}_{5}}\left( x,y \right)$ and studied certain properties, as well. Following Horn's method, we determine all possible cases of the convergence region of the function ${{D}_{1}}\left( x,y \right).$ Further, for a generalized hypergeometric function, ${{D}_{1}}\left( x,y \right)$ (two variable Mittag-Leffler-type function) integral representations of the Euler type have been proved. One-dimensional and two-dimensional Laplace transforms of the function are also defined. We have constructed a system of partial differential equations which is linked with the function ${{D}_{1}}\left( x,y \right)$.