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A product formula for multivariate Rogers-Szegö polynomials

Let $H_n(t)$ denote the classical Rogers-Szegö polynomial, and let $\tH_n(t_1, \ldots, t_l)$ denote the homogeneous Rogers-Szegö 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-Szegö 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.