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.

arXiv comment17 pages

References in corpus (5)

• The Prabhakar or three parameter Mittag--Leffler function: theory and application
• On a generalized three-parameter Wright function of the Le Roy type
• Sharp inequalities for the psi function and harmonic numbers
• Truncated generalized coherent states
• Differentiation of the Wright functions with respect to parameters and other results