On generalized Mittag-Leffler-type functions of two variables

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 ${{D}_{1}}\left( x,y \right)$.