Fractional operators and special functions. I. Bessel functions

TL;DR

This paper introduces fractional generalizations of Lie algebra operators, specifically for the Euclidean group E(2) and Bessel functions, revealing new integral representations and deepening the understanding of their group-theoretic properties.

Contribution

It defines and explores fractional operators within Lie theory, applying them to Bessel functions to derive new relations and integral representations.

Findings

Derived integral representations for Bessel functions using fractional operators

Reproduced known fractional integrals in a two-variable context

Linked properties of Bessel functions to underlying Lie group structure

Abstract

Most of the special functions of mathematical physics are connected with the representation of Lie groups. The action of elements $D$ of the associated Lie algebras as linear differential operators gives relations among the functions in a class, for example, their differential recurrence relations. In this paper, we define fractional generalizations $D^\mu$ of these operators in the context of Lie theory, determine their formal properties, and illustrate their use in obtaining interesting relations among the functions. We restrict our attention here to the Euclidean group E(2) and the Bessel functions. We show that the two-variable fractional operator relations lead directly to integral representations for the Bessel functions, reproduce known fractional integrals for those functions when reduced to one variable, and contribute to a coherent understanding of the connection of many properties of the functions to the underlying group structure. We extend the analysis to the associated Legendre functions in a following paper.