On generalized mittag-leffler-type functions of two variables

We aim to study Mittag-Leffler type functions of two variables D 1 ( x , y ) , . . . , D 5 ( x , y ) by analogy with the Appell hypergeometric functions of two variables,. Moreover, we targeted functions E 1 ( x , y ) , . . . , E 10 ( x , y ) as limiting cases of the functions D 1 ( x , y ) , . . . , D 5 ( x , y ) and studied certain properties, as well. Following Horn’s method, we determine all possible cases of the convergence region of the function D 1 ( x , y ) . Further, for a generalized hypergeometric function D 1 ( x , y ) (Mittag-Leffler type function) integral representations of the Euler type are 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 ( x , y ) .