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.

A practical guide to Prabhakar fractional calculus

Federico Polito;
2020-01-01

Abstract

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.
2020
23
1
9
54
https://arxiv.org/pdf/2002.10978
Andrea Giusti, Ivano Colombaro, Roberto Garra, Roberto Garrappa, Federico Polito, Marina Popolizio, Francesco Mainardi
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/1731084
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