Mittag-Leffler type functions of three variables

In this article, we generalized Mittag-Leffler-type functions ${\overline{F}}_A^{\left(3\right)},{\overline{F}}_B^{\left(3\right)},{\overline{F}}_C^{\left(3\right)}$, and ${\overline{F}}_D^{\left(3\right)}$, which correspond, respectively, to the familiar Lauricella hypergeometric functions $F_A^{\left(3\right)},F_B^{\left(3\right)},F_C^{\left(3\right)}$, and $F_D^{\left(3\right)}$ of three variables. Initially, from the Mittag-Leffler type function in the simplest form to the functions we are studying, necessary information about the development history, study, and importance of this and hypergeometric type functions will be introduced. Among the various properties and characteristics of these three-variable Mittag-Leffler-type function ${\overline{F}}_D^{\left(3\right)}$, which we investigate in the 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, and their connections with the Riemann-Liouville operators of fractional calculus. The link of three-variable Mittag-Leffler function with fractional differential equation systems involving different fractional orders is necessary on certain applications in physics.Therefore, we provide the systems of partial differential equations which are associated with them.