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

TL;DR

This paper generalizes the classical product formula for Rogers-Szeg"o polynomials to multivariate homogeneous versions, providing recursive and closed-form coefficients and connecting them to symmetric function theory.

Contribution

It introduces a new product formula for multivariate Rogers-Szeg"o polynomials with recursive and closed-form coefficients, extending classical results.

Findings

Derived a product formula for multivariate Rogers-Szeg"o polynomials

Provided recursive and closed-form expressions for coefficients

Connected coefficients to symmetric function structure constants

Abstract

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.