Realizations of su(1,1) and Uq(su(1,1)) and generating functions for orthogonal polynomials

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Abstract
Positive discrete series representations of the Lie algebra su(1,1) and the quantum algebra Uq(su(1,1)) are considered. The diagonalization of a self-adjoint operator (the Hamiltonian) in these representations and in tensor products of such representations is determined, and the generalized eigenvectors are constructed in terms of orthogonal polynomials. Using simple realizations of su(1,1), Uq(su(1,1)), and their representations, these generalized eigenvectors are shown to coincide with generating functions for orthogonal polynomials. The relations valid in the tensor product representations then give rise to new generating functions for orthogonal polynomials, or to Poisson kernels. In particular, a group theoretical derivation of the Poisson kernel for Meixner–Pollaczek and Al-Salam–Chihara polynomials is obtained.
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