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

Abstract

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 |x₁|^α₁ . . . |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.