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

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.