  Positive discrete series representations of the Lie algebra $su(1,1)$ and the
quantum algebra $U_q(su(1,1))$ 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 $su(1,1)$, $U_q(su(1,1))$, 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-Pollaczak and Al-Salam--Chihara
polynomials is obtained.
