  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.
