Elementary observations on Rogers-Szeg\"o polynomials

TL;DR

This paper explores special cases of Rogers-Szeg"o polynomials, revealing closed-form expressions and properties related to their expansions and determinants, with an elementary and accessible presentation.

Contribution

It identifies exceptional cases of Rogers-Szeg"o polynomials with closed formulas and analyzes their properties, providing new insights into their structure.

Findings

Certain Rogers-Szeg"o polynomials have closed-form expressions.

Normalized Rogers-Szeg"o polynomials exhibit nice expansions.

Hankel determinants associated with these polynomials have notable properties.

Abstract

The Rogers-Szeg\"o polynomials are natural q-analogues of Newton binomials. In general they have no closed expression. We consider some exceptional cases which are products of a factor with a closed formula and another one with nice values for q=1 and q=-1. These are related to normalized Rogers-Szeg\"o polynomials which have nice expansions and Hankel determinants. The Exposition of the paper is elementary and almost self-contained.