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.