MULTIVARIABLE ORTHOGONAL POLYNOMIALS AND COUPLING-COEFFICIENTS FOR DISCRETE-SERIES REPRESENTATIONS

We study polynomials of several variables which occur as coupling coef ficients for the analytic continuation of the holomorphic discrete ser ies of SU(1, 1). There are three types of such polynomials, one corres ponding to each conjugacy class of one-parameter subgroups. They may b e viewed as multivariable generalizations of Hahn, Jacobi, and continu ous Hahn polynomials and include many orthogonal and biorthogonal fami lies occurring in the literature. We give a simple and unified approac h to these polynomials using the group theoretic interpretation. We pr ove many formal properties, in particular a number of convolution and linearization formulas. We also develop the corresponding theory for t he Heisenberg group, leading to multivariable generalizations of Krawt chouk and Hermite polynomials.