A Class of Generalized Mittag-Leffler-Type Functions Associated with the Lauricella Functions of Three Variables

Abstract

In this article, we aim to study the Mittag-Leffler-type functions
$\widetilde{F}_{A}^{\left(3\right)}$, $\widetilde{F}_{B}^{\left(3\right)}$,
$\widetilde{F}_{C}^{\left(3\right)}$ and $\widetilde{F}_{D}^{\left(3\right)}$,
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 functions, 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,
the systems of partial differential equations which
are are satisfied by them, their Euler-type integral
representations, their one- as well as three-dimensional Laplace
transforms, and their connections with the Riemann-Liouville
operators of fractional calculus.