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

We work out the expression of the generalized Bessel function of B₂-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 B₂-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].