Abstract
Dedicated to the memory of Professor Richard Askey (1933–2019) and to pay tribute to the Bateman Project. Harry Bateman planned his “shoe-boxes” project (accomplished after his death as Higher Transcendental Functions, Vols. 1–3, 1953–1955, under the editorship by A. Erdélyi) as a “Guide to the Functions”. This inspired the author to use the modified title of the present survey. Most of the standard (classical) Special Functions are representable in terms of the Meijer G-function and, specially, of the generalized hypergeometric functions pFq. These appeared as solutions of differential equations in mathematical physics and other applied sciences that are of integer order, usually of second order. However, recently, mathematical models of fractional order are preferred because they reflect more adequately the nature and various social events, and these needs attracted attention to “new” classes of special functions as their solutions, the so-called Special Functions of Fractional Calculus (SF of FC). Generally, under this notion, we have in mind the Fox H-functions, their most widely used cases of the Wright generalized hypergeometric functions pΨq and, in particular, the Mittag–Leffler type functions, among them the “Queen function of fractional calculus”, the Mittag–Leffler function. These fractional indices/parameters extensions of the classical special functions became an unavoidable tool when fractalized models of phenomena and events are treated. Here, we try to review some of the basic results on the theory of the SF of FC, obtained in the author’s works for more than 30 years, and support the wide spreading and important role of these functions by several examples.
Highlights
Dedicated to the memory of Professor Richard Askey (1933–2019) and to pay tribute to the Bateman Project
In the book [9] and subsequent papers, we provided a full set of operational properties of the operators (23) and (24) that justify their names as operators of Generalized Fractional Calculus (GFC), as semigroup property, formal inversion formula, reduction to identity or to the conventional integration operators for special parameters’ choice
Under the notion of Special Functions of Fractional Calculus (SF of FC), we have in mind the Fox H-function and the Wright generalized hypergeometric functions p Ψq, including the Mittag–Leffler function, its multi-index extensions and all their particular cases
Summary
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, astronomy, statistics or other applications Algebra Systems packages Mathematica and Maple are based, the Bateman Project [3] and the NIST Project [4] based on the Abramowitz–Stegun handbook [5] and on a more recent one, edited by Olver–Lozier–Boisvert–Clark [6] The author of this survey was tempted to start paying attention to Special Functions by having the handbook [3] on her desk, while working on a M.Sc. thesis. During his last years he (Professor Harry Bateman) had embarked upon a project whose successful completion, he believed, would prove of great value to scientists in all fields He planned an extensive compilation of “special functions”—solutions of a wide class of mathematically and physically relevant functional equations. Mourad Ismail and Walter Van Assche [8]
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