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

TL;DR

This paper rigorously justifies the term-by-term differentiation of Mittag-Leffler type functions with respect to their parameters using uniform convergence, covering various parametric forms including Prabhakar, Le Roy, and Wright functions.

Contribution

It provides a formal proof for differentiation formulas of Mittag-Leffler functions with multiple parameters, expanding the theoretical foundation for their parametric analysis.

Findings

Legitimacy of parameter differentiation for Mittag-Leffler functions established

Differentiation formulas justified via uniform convergence

Applicable to functions represented by Mellin-Barnes integrals

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.