# Coupling coefficients of $su_q(1,1)$ and multivariate $q$-Racah polynomials

## TL;DR

This paper links multivariate $q$-Racah polynomials to quantum algebra representations and $q$-deformed systems, revealing their role as connection coefficients and $3nj$ symbols in a novel mathematical framework.

## Contribution

It demonstrates that multivariate $q$-Racah polynomials can be interpreted as connection coefficients and $3nj$ symbols, connecting quantum algebra representations with $q$-deformed physical systems.

## Key findings

- Multivariate $q$-Racah polynomials arise as connection coefficients.
- $q$-Hahn polynomials are constructed via Clebsch--Gordan coefficients.
- $q$-Hahn polynomials appear in wavefunctions of $q$-deformed systems.

## Abstract

Gasper & Rahman's multivariate $q$-Racah polynomials are shown to arise as connection coefficients between families of multivariate $q$-Hahn or $q$-Jacobi polynomials. The families of $q$-Hahn polynomials are constructed as nested Clebsch--Gordan coefficients for the positive-discrete series representations of the quantum algebra $su_q(1,1)$. This gives an interpretation of the multivariate $q$-Racah polynomials in terms of $3nj$ symbols. It is shown that the families of $q$-Hahn polynomials also arise in wavefunctions of $q$-deformed quantum Calogero--Gaudin superintegrable systems.