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

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

2009-05-14

arxiv.org/abs/0905.2265v1

Mathematics

Probability

this paper is accepted for publication in Int. Trans. Spec. Funct

Scientific paper

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.

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