
Journal of Lie Theory 22 (2012), No. 1, 081091 Copyright Heldermann Verlag 2012 Generalized Bessel Function Associated with Dihedral Groups Nizar Demni IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes, France nizar.demni@univrennes1.fr 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 positive integer. As a byproduct, we get a relatively simple integral representation for the generalized Bessel function associated with dihedral groups D_{n}, n ≥ 2 when both multiplicities sum to an integer. In particular, we recover a previous result obtained for D_{4} and we give a special interest to D_{6}. Finally, we derive similar results for odd dihedral groups. Keywords: Generalized Bessel function, dihedral groups, Jacobi polynomials, Radon Transform. MSC: 33C52, 33C45, 42C10, 43A85, 43A90 [ Fulltextpdf (285 KB)] for subscribers only. 