Multivariable Orthogonal Polynomials and Coupling Coefficients for Discrete Series Representations

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We study polynomials of several variables which occur as coupling coefficients for the analytic continuation of the holomorphic discrete series of SU(1,1). There are three types of such polynomials, one corresponding to each conjugacy class of one-parameter subgroups. They may be viewed as multivariable generalizations of Hahn, Jacobi, and continuous Hahn polynomials and include many orthogonal and biorthogonal families occurring in the literature. We give a simple and unified approach to these polynomials using the group theoretic interpretation. We prove many formal properties, in particular a number of convolution and linearization formulas. We also develop the corresponding theory for the Heisenberg group, leading to multivariable generalizations of Krawtchouk and Hermite polynomials.