Product formula for Jacobi polynomials, spherical harmonics and generalized Bessel function of dihedral type

TL;DR

This paper derives explicit integral formulas for the generalized Bessel function of type B in two dimensions, using Jacobi polynomial product formulas, advancing the understanding of Dunkl operators and their actions on symmetric functions.

Contribution

It provides a new explicit integral expression for the generalized Bessel function and the Dunkl V operator action for dihedral groups, extending previous results.

Findings

Explicit integral formula for the generalized Bessel function of type B.

Extended the V operator action to all dihedral systems.

Improved upon previous formulas by Y. Xu.

Abstract

We work out the expression of the generalized Bessel function of type B in the two-rank case. This is done using Dijskma and Koornwinder's product formula for Jacobi polynomials 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 toward the explicit expression of the Dunkl's intertwining V operator reflections-invariant functions. Finally, we give in the same setting an explicit formula for the action of V on a product of a power of the norm and a spherical harmonic. The obtained formula extends to all dihedral systems and it improves the one derived by Y.Xu.