Product Formulas for a Two-Parameter Family of Heckman-Opdam Hypergeometric Functions of Type BC

摘要

In this paper we present explicit product formulas for a continuous two-parameter family of Heckman-Opdam hypergeometric functions of type BC on Weyl chambers C-q subset of R-q of type B. These formulas are related to continuous one-parameter families of probability-preserving convolution structures on C-q x R. These convolutions on C-q x R are constructed via product formulas for the spherical functions of the symmetric spaces U(p,q)/(U(p) x SU(q)) and associated double coset convolutions on C-q x T with the torus T. We shall obtain positive product formulas for a restricted parameter set only, while the associated convolutions are always norm-decreasing.