Title: On fractional powers of Bessel operators,
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.