## Abstract

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 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.

Original language | English (US) |
---|---|

Pages (from-to) | 81-91 |

Number of pages | 11 |

Journal | Journal of Lie Theory |

Volume | 22 |

Issue number | 1 |

State | Published - 2012 |

## Keywords

- Dihedral groups
- Generalized Bessel function
- Jacobi polynomials
- Radon Transform.

## ASJC Scopus subject areas

- Algebra and Number Theory