Rogers–Szegő polynomials

Not to be confused with Rogers 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

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

↑ Vinroot, C. Ryan (12 July 2012). "An enumeration of flags in finite vector spaces". The Electronic Journal of Combinatorics. 19 (3).

Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, ISBN 978-0-521-83357-8, MR 2128719
Szegő, Gábor (1926), "Beitrag zur theorie der thetafunktionen", Sitz Preuss. Akad. Wiss. Phys. Math. Ki., XIX: 242–252, Reprinted in his collected papers

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