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

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 $s{u}_q(1,1)$. 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.