Journal of Lie Theory 22 (2012), No. 1, 081--091
Copyright Heldermann Verlag 2012
Generalized Bessel Function Associated with Dihedral Groups
IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes, France
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 by-product, we get a relatively simple integral representation for the generalized Bessel function associated with dihedral groups Dn, n ≥ 2 when both multiplicities sum to an integer. In particular, we recover a previous result obtained for D4 and we give a special interest to D6. 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
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