Rogers–Szegő polynomials

Template:Distinguish Template:Format footnotes In mathematics, the Rogers–Szegő polynomials are a family of polynomials orthogonal on the unit circle introduced by Template:Harvs, 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 $n \geq 1$ ) the recurrence relation^[1]

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 $h_{1} (x; q) = 1 + x$ .

References