On a Neumann-type series for modified Bessel functions of the first kind

TL;DR

This paper investigates a Neumann-type series for modified Bessel functions linked to Dunkl operators and dihedral groups, providing new proofs and representations that deepen understanding of their structure and connections to hypergeometric functions.

Contribution

It offers new proofs and representations of a Neumann-type series for Bessel functions, including cases related to dihedral groups and hypergeometric functions, expanding theoretical understanding.

Findings

New proofs for the series in special cases.

Representation of the series as a $\

A new expression for Gegenbauer polynomials.

Abstract

In this paper, we are interested in a Neumann-type series for modified Bessel functions of the first kind which arises in the study of Dunkl operators associated with dihedral groups and as an instance of the Laguerre semigroup constructed by Ben Said-Kobayashi-Orsted. We first revisit the particular case corresponding to the group of square-preserving symmetries for which we give two new and different proofs other than the existing ones. The first proof uses the expansion of powers in a Neumann series of Bessel functions while the second one is based on a quadratic transformation for the Gauss hypergeometric function and opens the way to derive further expressions when the orders of the underlying dihedral groups are powers of two. More generally, we give another proof of De Bie \& al formula expressing this series as a $\Phi_2$-Horn confluent hypergeometric function. In the course of proving, we shed the light on the occurrence of multiple angles in their formula through elementary symmetric functions, and get a new representation of Gegenbauer polynomials.