On differentiation with respect to parameters of the functions of the Mittag-Leffler type

Abstract

The formal term-by-term differentiation with respect to parameters is demonstrated to be legitimate for the Mittag-Leffler type functions. The justification of differentiation formulas is made by using the concept of the uniform convergence. This approach is applied to the Mittag-Leffler function depending on two parameters and, additionally, for the $3$-parametric Mittag-Leffler functions (namely, for the Prabhakar function and the Le Roy type functions), as well as for the $4$-parametric Mittag-Leffler function (and, in particular, for the Wright function). The differentiation with respect to the involved parameters is discussed also in case those special functions which are represented via the Mellin-Barnes integrals.