Coupling coefficients for tensor product representations of quantum $\mathrm{SU}(2)$

TL;DR

This paper investigates tensor product representations of the quantum group SU(2), deriving coupling coefficients that serve as q-analogs of Bessel functions, and establishing related q-integral identities involving q-hypergeometric polynomials.

Contribution

It introduces explicit coupling coefficients for infinite dimensional tensor product representations of SU(2) quantum group as q-analogs of Bessel functions, along with new q-integral identities.

Findings

Coupling coefficients are expressed as q-analogs of Bessel functions.

Derived q-integral identities involving q-hypergeometric polynomials.

Eigenvectors of certain self-adjoint elements are characterized.

Abstract

We study tensor products of infinite dimensional representations (not corepresentations) of the $\mathrm{SU}(2)$ quantum group. Eigenvectors of certain self-adjoint elements are obtained, and coupling coefficients between different eigenvectors are computed. The coupling coefficients can be considered as $q$-analogs of Bessel functions. As a results we obtain several $q$-integral identities involving $q$-hypergeometric orthogonal polynomials and $q$-Bessel-type functions.