Orthogonal Polynomials for a Family of Product Weight Functions on the Spheres

Based on the theory of spherical harmonics for measures invariant under a finite reflection group developed by Dunkl recently, we study orthogonal polynomials with respect to the weight functions | x 1 | α 1 . . . | x d | α d on the unit sphere S d -1 in ℝ d . The results include explicit formulae for orthonormal polynomials, reproducing and Poisson kernel, as well as intertwining operator.