A practical guide to Prabhakar fractional calculus

The Mittag–Leffler function is universally acclaimed as the Queen function of fractional calculus. The aim of this work is to survey the key results and applications emerging from the three-parameter generalization of this function, known as the Prabhakar function. Specifically, after reviewing key historical events that led to the discovery and modern development of this peculiar function, we discuss how the latter allows one to introduce an enhanced scheme for fractional calculus. Then, we summarize the progress in the application of this new general framework to physics and renewal processes. We also provide a collection of results on the numerical evaluation of the Prabhakar function.

Differential and integral relations in the class of multi-index Mittag-Leffler functions

As recently observed by Bazhlekova and Dimovski [1], the n-th derivative of the 2-parametric Mittag-Leffler function gives a 3-parametric Mittag-Leffler function, known as the Prabhakar function. Following this analogy, the n-th derivative of the (2m-index) multi-index Mittag-Leffler functions [6] is obtained, and it turns out that it is expressed in terms of the (3m-index) Mittag-Leffler functions [10, 11]. Further, some special cases of the fractional order Riemann-Liouville and Erdélyi-Kober integrals of the Mittag-Leffler functions are calculated and interesting relations are proved. Analogous relations happen to connect the 3m-Mittag-Leffler functions with the integrals and derivatives of 2m-Mittag-Leffler functions. Finally, multiple Erdélyi-Kober fractional integration operators, as operators of the generalized fractional calculus [5], are shown to relate the 2m- and 3m-parametric Mittag-Leffler functions.

On Euler type integrals involving extended Mittag-Leffler functions

The Object of the present paper is to establish some interested theorems on Euler type integral involving extended Mittag-Leffler function. Further, we reduce some special cases involving various known functions like Wiman function, Prabhakar function, exponential and Binomial functions.