From the hyper-Bessel operators of Dimovski to the generalized fractional calculus

Abstract

In 1966 Ivan Dimovski introduced and started detailed studies on the Bessel type differential operators B of arbitrary (integer) order m ≥ 1. He also suggested a variant of the Obrechkoff integral transform (arising in a paper of 1958 by another Bulgarian mathematician Nikola Obrechkoff) as a Laplace-type transform basis of a corresponding operational calculus for B and for its linear right inverse integral operator L. Later, the developments on these linear singular differential operators appearing in many problems of mathematical physics, have been continued by the author of this survey who called them hyper-Bessel differential operators, in relation to the notion of hyper-Bessel functions of Delerue (1953), shown to form a fundamental system of solutions of the IVPs for By(t) = λy(t). We have been able to extend Dimovski's results on the hyper-Bessel operators and on the Obrechkoff transform due to the happy hint to attract the tools of the special functions as Meijer's G-function and Fox's H-function to handle successfully these matters. These author's studies have lead to the introduction and development of a theory of generalized fractional calculus (GFC) in her monograph (1994) and subsequent papers, and to various applications of this GFC in other topics of analysis, differential equations, special functions and integral transforms. Here we try briefly to expose the ideas leading to this GFC, its basic facts and some of the mentioned applications.MSC . Definitions of classical FCThe classical fractional calculus can be thought as an upgrade of the Calculus, originated as early as in 1695 (the famous l'Hospital's letter to Leibnitz). Since then, many known mathematicians and applied scientists contributed to the development of this "strange" calculus, but the first book and the first conference dedicated specially to that topic appeared 279 years (1974−1695 = 279) after the mentioned correspondence. And this year we are marking 40 years (2014−1974 = 40) of these two events, when there are published more than 100 monographs and topical selections on the area of FC and its applications. The detailed history, theory and its various applications, by the years of 1987-1993 was presented in the "FC Encyclopedia" [42], and currently -in several recent surveys as [47], and the posters at http://www.math.bas.bg/∼fcaa.The classical FC is based on several (equivalent or alternative) definitions for the operators of integration and differentiation of arbitrary (including real fractional or complex) order, as continuations of the classical integration and differentiation operators and their integer order powers (n ∈ N), namely -the n-fold integration

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