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

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

2010-09-29

arxiv.org/abs/1009.5797v1

Mathematics

Classical Analysis and ODEs

Submitted

Scientific paper

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.

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