Rogers–Szegő polynomials

In mathematics, the Rogers–Szegő polynomials are a family of polynomials orthogonal on the unit circle introduced by Szegő (1926), who was inspired by the continuous q-Hermite polynomials studied by Leonard James Rogers. They are given by

${\displaystyle h_{n}(x;q)=\sum _{k=0}^{n}{\frac {(q;q)_{n}}{(q;q)_{k}(q;q)_{n-k}}}x^{k}}$

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

Furthermore, the ${\displaystyle h_{n}(x;q)}$ satisfy (for ${\displaystyle n\geq 1}$) the recurrence relation[1]

${\displaystyle h_{n+1}(x;q)=(1+x)h_{n}(x;q)+x(q^{n}-1)h_{n-1}(x;q)}$

with ${\displaystyle h_{0}(x;q)=1}$ and ${\displaystyle h_{1}(x;q)=1+x}$.