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Journal of Lie Theory 22 (2012), No. 1, 081--091
Copyright Heldermann Verlag 2012

Generalized Bessel Function Associated with Dihedral Groups

Nizar Demni
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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