A product formula for multivariate Rogers-Szeg\"o polynomials

Let $H_n(t)$ denote the classical Rogers-Szeg\"o polynomial, and let $\tH_n(t_1, \ldots, t_l)$ denote the homogeneous Rogers-Szeg\"o polynomial in $l$ variables, with indeterminate $q$. There is a classical product formula for $H_k(t)H_n(t)$ as a sum of Rogers-Szeg\"o polynomials with coefficients being polynomials in $q$. We generalize this to a product formula for the multivariate homogeneous polynomials $\tH_n(t_1, \ldots, t_l)$. The coefficients given in the product formula are polynomials in $q$ which are defined recursively, and we find closed formulas for several interesting cases. We then reinterpret the product formula in terms of symmetric function theory, where these coefficients become structure constants.