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

Abstract

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.

20 pages, LaTeX2e, to appear in J. Math. Phys

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• Finite q-oscillator
• Group Theory and Quasiprobability Integrals of Wigner Functions
• A finite oscillator model related to sl(2|1)
• Discrete series representations for sl(2|1), Meixner polynomials and oscillator models
• Basic Hypergeometric Functions and Covariant Spaces for Even Dimensional Representations of U_q[osp(1/2)]
• Wigner quantization of some one-dimensional Hamiltonians
• Exponential generating functions for the associated Bessel functions
• Convolution identities for Dunkl orthogonal polynomials from the $\mathfrak{osp}(1|2)$ Lie superalgebra