A Practical Guide to Prabhakar Fractional Calculus

• Published in: Fractional calculus & applied analysis Vol. 23; no. 1; pp. 9 - 54
• Main Authors: Giusti, Andrea, Colombaro, Ivano, Garra, Roberto, Garrappa, Roberto, Polito, Federico, Popolizio, Marina, Mainardi, Francesco
• Format: Journal Article
• Language: English
• Published: Warsaw Versita 01.02.2020; De Gruyter; Nature Publishing Group
• Subjects: 26A33; 34K37; 60G22; Abstract Harmonic Analysis; Analysis; anomalous physical phenomena; Calculus; Fractional calculus; Functional Analysis; Integral Transforms; Mathematics; Mittag-Leffler type functions; numerical methods; Operational Calculus; Prabhakar fractional calculus; Prabhakar function; Primary: 33E12; Secondary: 65R10; stochastic processes; Survey Paper; Prabhakar function; numerical methods; Primary: 33E12, 26A33; Prabhakar fractional calculus; Mittag-Leffler type functions; stochastic processes; Secondary: 65R10, 34K37, 60G22; anomalous physical phenomena
• ISSN: 1311-0454; 1314-2224
• DOI: 10.1515/fca-2020-0002

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.