Radon Transform on spheres and generalized Bessel function associated with dihedral groups

TL;DR

This paper explores the Radon transform and generalized Bessel functions related to dihedral groups, providing new formulas and integral representations that extend previous results and apply to all Weyl dihedral groups.

Contribution

It introduces a unified approach to derive formulas for generalized Bessel functions associated with dihedral groups using Fourier and Radon transforms.

Findings

Derived a closed formula for a series involving Bessel and Gegenbauer functions.

Provided integral representations for generalized Bessel functions for even dihedral groups.

Extended methods to odd dihedral groups, covering all Weyl dihedral groups.

Abstract

Motivated by Dunkl operators theory, we consider a generating series involving a modified Bessel function and a Gegenbauer polynomial, that generalizes a known series already considered by L. Gegenbauer. We actually use inversion formulas for Fourier and Radon transforms to derive a closed formula for this series when the parameter of the Gegenbauer polynomial is a strictly positive integer. As a by-product, we get a relatively simple integral representation for the generalized Bessel function associated with even dihedral groups when both multiplicities sum to an integer. In particular, we recover a previous result obtained for the square symmetries-preserving group and we give a special interest to the hexagon. The paper is closed with adapting our method to odd dihedral groups thereby exhausting the list of Weyl dihedral groups.