  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.
