A Short Essay on the Special Functions of Fractional Calculus

This short announce is based on the author’s recent publications on the so-called Special Functions of Fractional Calculus (SF of FC)—some of them listed below, as well as on a series of our invited lectures at the Summer School “Singularities in Science and Engineering” in Ghent University, 22–31 August 2022, hosted by the “Analysis and PDE” group there. Under SF of FC, we mean in general a wide variety of generalized hypergeometric functions as Fox’ H-functions, and specially their particular cases—the Wright generalized hypergeometric functions \({ }_p\Psi _q\) , the Mittag-Leffler functions \(E_{\alpha ,\beta }\) and their multi-index and multi-variable analogues. These are unavoidable tools as explicit analytical solutions of fractional order differential and integral equations and systems that model various phenomena in both natural and social sciences. We note that the elementary functions and the “classical SF” or “SF of Mathematical Physics” (including also orthogonal polynomials) are part of the scheme of SF of FC, as solutions of integer order (mainly 2nd order) mathematical models, and most of them can be represented by the simpler generalized hypergeometric functions as \({ }_pF_q\) and Meijer’s G-functions.