A Brief Story about the Operators of the Generalized Fractional Calculus

2000 Mathematics Subject Classification: 26A33, 33C60, 44A20In this survey we present a brief history and the basic ideas of the generalized fractional calculus (GFC). The notion “generalized operator of fractional integration” appeared in the papers of the jubilarian Prof. S.L. Kalla in the years 1969-1979 when he suggested the general form of these operators and studied examples of them whose kernels were special functions as the Gauss and generalized hypergeometric functions, including arbitrary G- and H-functions. His ideas provoked the author to choose a more peculiar case of such kernels and to develop a theory of the corresponding GFC that featured many applications. All known fractional integrals and derivatives and other generalized integration and differential operators in various areas of analysis happened to fall in the scheme of this GFC

On Two Saigo’s Fractional Integral Operators in the Class of Univalent Functions

2000 Mathematics Subject Classification: Primary 26A33, 30C45; Secondary 33A35Recently, many papers in the theory of univalent functions have been devoted to mapping and characterization properties of various linear integral or integro-differential operators in the class S (of normalized analytic and univalent functions in the open unit disk U), and in its subclasses (as the classes S∗ of the starlike functions and K of the convex functions in U). Among these operators, two operators introduced by Saigo, one involving the Gauss hypergeometric function, and the other - the Appell (or Horn) F3-function, are rather popular. Here we view on these Saigo’s operators as cases of generalized fractional integration operators, and show that the techniques of the generalized fractional calculus and special functions are helpful to obtain explicit sufficient conditions that guarantee mappings as: S → S and K → S, that is, preserving the univalency of functions.* Partially supported by National Science Fund (Bulg. Ministry of Educ. and Sci.) under Project MM 1305

A Guide to Special Functions in Fractional Calculus

Dedicated to the memory of Professor Richard Askey (1933-2019) and to pay tribute to the Bateman Project. Harry Bateman planned his project (accomplished after his death as Higher Transcendental Functions, Vols. 1-3, 1953-1955, under the editorship by A. Erdelyi) 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

A Poster about the Recent History of Fractional Calculus

MSC 2010: 26A33, 05C72, 33E12, 34A08, 34K37, 35R11, 60G22In the last decades fractional calculus became an area of intense re-search and development. The accompanying poster illustrates the major contributions during the period 1966-2010

A Poster about the Old History of Fractional Calculus

MSC 2010: 26A33, 05C72, 33E12, 34A08, 34K37, 35R11, 60G22The fractional calculus (FC) is an area of intensive research and development. In a previous paper and poster we tried to exhibit its recent state, surveying the period of 1966-2010. The poster accompanying the present note illustrates the major contributions during the period 1695-1970, the "old history" of FC

Recent history of fractional calculus

This survey intends to report some of the major documents and events in the area of fractional calculus that took place since 1974 up to the present date

Trends, directions for further research, and some open problems of fractional calculus

The area of fractional calculus (FC) has been fast developing and is presently being applied in all scientific fields. Therefore, it is of key relevance to assess the present state of development and to foresee, if possible, the future evolution, or, at least, the challenges identified in the scope of advanced research works. This paper gives a vision about the directions for further research as well as some open problems of FC. A number of topics in mathematics, numerical algorithms and physics are analyzed, giving a systematic perspective for future research.info:eu-repo/semantics/publishedVersio

Contributions to Round Table Discussion held at ICFDA 2016

Along with the presentations made during this Round Table, we include here some contributions by the participants sent afterwards and also by few colleagues planning but failed to attend. The intention of these discussions was to continue the useful traditions from the first conferences on Fractional Calculus (FC) topics, to pose open problems, challenging hypotheses and questions “where to go”, “how to save and improve the prestige of FC”, to share opinions and try to find ways to resolve them.info:eu-repo/semantics/publishedVersio