Rogers–Szegő polynomials

In mathematics, the Rogers–Szegő polynomials are a family of polynomials orthogonal on the unit circle introduced by, who was inspired by the continuous q-Hermite polynomials studied by Leonard James Rogers. They are given by
 * $$h_n(x;q) = \sum_{k=0}^n\frac{(q;q)_n}{(q;q)_k(q;q)_{n-k}}x^k$$

where (q;q)n is the descending q-Pochhammer symbol.

Furthermore, the $$h_n(x;q)$$ satisfy (for $$n \ge 1$$) the recurrence relation
 * $$ h_{n+1}(x;q) = (1+x)h_n(x;q) + x(q^n-1)h_{n-1}(x;q)$$

with $$h_0(x;q)=1$$ and $$h_1(x;q)=1+x$$.