Product Formula for Jacobi Polynomials, Spherical Harmonics and Generalized Bessel Function of Dihedral Type

Abstract. We work out the expression of the generalized Bessel function of B2-type derived in [4]. 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 operator Vk in the B2invariant setting. Finally, we give in the same setting an explicit formula for the action of Vk on a product of |y| , κ ≥ 0 and the ordinary spherical harmonic Y4m(y) := |y| cos(4mθ), y = |y|e. The obtained formula extends to all dihedral systems and it improves the one derived in [16].