Multivariate Askey-Wilson functions and overlap coefficients

We study certain overlap coefficients appearing in representation theory of the quantum algebra ${\text{U}}_q({\mathfrak{s}\mathfrak{l}}_2(\text{C}))$. The overlap coefficients can be identified as products of Askey-Wilson functions, leading to an algebraic interpretation of the multivariate Askey-Wilson functions introduced by Geronimo and Iliev. We use the underlying coalgebra structure to derive $q$-difference equations satisfied by the multivariate Askey-Wilson functions.