## Abstract

We work out the expression of the generalized Bessel function of B_{2}-type derived in [N. Demni, Radial Dunkl processes associated with Dihedral systems. Séminaire de Probabilités XLII]. This is done using Dijksma and Koornwinder's product formula for Jacobi polynomials [A. Dijksma and T.H. Koornwinder, Spherical Harmonics and the product of two Jacobi polynomials, Indag. Math. 33 (1971), pp. 191-196], and the obtained expression is given by multiple integrals involving only a normalized modified Bessel function and two symmetric Beta distributions. We think of that expression as the major step towards the explicit expression of the Dunkl's intertwining operator V_{k} in the B_{2}-invariant setting. Finally, we give in the same setting an explicit formula for the action of V_{k} on a product of {pipe}y{pipe}^{2κ}, κ ≥ 0, and the ordinary spherical harmonic Y_{4m}(y) := {pipe}y{pipe}^{4m} cos(4mθ), y = {pipe}y{pipe}e^{iθ}. The obtained formula extends to all dihedral systems and it improves the one derived in [Y. Xu, Intertwining operator and h-harmonics associated with reflection group, Can. J. Math. 50 (1998), pp. 193-209].

Original language | English (US) |
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Pages (from-to) | 105-123 |

Number of pages | 19 |

Journal | Integral Transforms and Special Functions |

Volume | 21 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2010 |

## Keywords

- Dunkl's intertwining operator
- Gegenbauer polynomials
- Generalized Bessel function
- Jacobi polynomials
- Modified Bessel function

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics