On fractional powers of Bessel operators

TL;DR

This paper investigates fractional powers of the Bessel differential operator, providing explicit integral definitions, exploring their properties, and connecting them with Mellin and Hankel transforms, while highlighting potential for further research.

Contribution

It introduces explicit integral definitions of fractional Bessel operators without transforms and studies their properties and relations, expanding understanding of their mathematical structure.

Findings

Derived explicit integral forms of fractional Bessel operators

Established relations with Mellin and Hankel transforms

Discussed properties like group property and generalized Taylor formula

Abstract

This paper was published in the special issue of the Journal of Inequalities and Special Functions dedicated to Professor Ivan Dimovski's contributions to different fields of mathematics: transmutation theory, special functions, integral transforms, function theory etc. In this paper we study fractional powers of the Bessel differential operator. The fractional powers are defined explicitly in the integral form without use of integral transforms in its definitions. Some general properties of the fractional powers of the Bessel differential operator are proved and some are listed. Among them are different variations of definitions, relations with the Mellin and Hankel transforms, group property, generalized Taylor formula with Bessel operators, evaluation of resolvent integral operator in terms of the Wright or generalized Mittag--Leffler functions. At the end, some topics are indicated for further study and possible generalizations. Also the aim of the paper is to attract attention and give references to not widely known results on fractional powers of the Bessel differential operator.