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

TL;DR

This paper explores the representations of $su(1,1)$ and $U_q(su(1,1))$, constructing eigenvectors linked to orthogonal polynomials and deriving new generating functions and Poisson kernels through group theoretical methods.

Contribution

It provides a novel group theoretical approach to derive generating functions and Poisson kernels for orthogonal polynomials using representations of $su(1,1)$ and its quantum algebra.

Findings

Constructed eigenvectors in terms of orthogonal polynomials.

Derived new generating functions and Poisson kernels for orthogonal polynomials.

Provided a group theoretical derivation of the Poisson kernel for specific polynomial families.

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.