Fractional differential equations for the generalized Mittag-Leffler function

Abstract

In this paper, we establish some (presumably new) differential equation formulas for the extended Mittag-Leffler-type function by using the Saigo-Maeda fractional differential operators involving the Appell function $F_{3}(\cdot)$ and results in terms of the Wright generalized hypergeometric-type function ${}_{m+1}\psi^{(\{\kappa_{l}\}_{l\in\mathbb{N}_{0}} )}_{n+1}(z; p)$ recently established by Agarwal. Some interesting special cases are also pointed out.