Orthogonal polynomials on the ball and the simplex for weight functions with reflection symmetries

Generalized classical orthogonal polynomials on the unit ball B-d and the s tandard simplex T-d are orthogonal with respect to weight functions that ar e reflection-invariant on B-d and, after a composition, on T-d, respectivel y. They are also eigenfunctions of a second-order differential-difference o perator that is closely related to Dunkl's h-laplacian for the reflection g roups. Under a proper limit, the generalized classical orthogonal polynomia ls on B-d converge to the generalized Hermite polynomials on R-d, and those on T-d converge to the generalized Laguerre polynomials on R-+(d). The lat ter two are related to the Calogero-Sutherland models associated to the Wey l groups of type A and type B.