Mittag-Leffler type functions of three variables

In this article, we generalized Mittag-Leffler-type functions F ̵̄ A ( 3 ) , F ̵̄ B ( 3 ) , F ̵̄ C ( 3 ) and F ̵̄ D ( 3 ) , which correspond, respectively, to the familiar Lauricella hypergeometric functions F A ( 3 ) , F B ( 3 ) , F C ( 3 ) and F D ( 3 ) of three variables. Among the various properties and characteristics of these three-variable Mittag-Leffler-type function F ̵̄ D ( 3 ) , which we investigate in this article, include their relationships with other extensions and generalizations of the classical Mittag-Leffler functions, their three-dimensional convergence regions, their Euler-type integral representations, their Laplace transforms, their connections with the Riemann-Liouville operators of fractional calculus, and the systems of partial differential equations which are associated with them.