On two classes of generalized fractional operators (with short historical survey of fractional calculus)

TL;DR

This survey explores key variants of fractional operators, their historical development, and recent advances by the authors, including Buschman-Erdelyi operators and fractional powers of Bessel operators, highlighting their significance in mathematical analysis.

Contribution

The paper provides a comprehensive survey of fractional operators, emphasizing the authors' recent work on Buschman-Erdelyi operators and fractional Bessel operators, with historical context and applications.

Findings

Detailed classification of fractional integrodifferential operators

Introduction of Buschman-Erdelyi operators and their properties

Representation of fractional powers of Bessel operators

Abstract

This is a survey paper in two parts. In the first part we list main variants of one-dimensional fractional integrodifferential operators. Also some historical and priority remarks are given. As a special question we consider the impact of Soviet and Russian researches to fractional calculus and its applications to viscoelasticity theory. We also stress an impact of the Voronezh school of mechanics and viscoelasticity theory, including works of Shermergor, Meschkov, Rossikhin, Shitikova and others. In the second part of the paper we consider two important special classes of generalized fractional operators, they were thoroughly studied by the authors. We consider Buschman-Erdelyi operators, this is an important class containing as special cases Riemann-Liouville, Erd\'elyi-Kober, Mehler-Fock operators and some others. They also include classical transmutations for the Bessel differential operator, namely Sonine and Poisson ones. After that we represent results on another important generalization of Riemann-Liouville operators, namely fractional powers of the Bessel operators. The paper is dedicated to Adam Maremovich Nakhushev -- brilliant man and mathematician.