Rogers–Szegő polynomials

Template:More citations neededTemplate:Distinguish 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 $n\ge 1$) 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 $h_1(x;q)=1+x$.

Further reading

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