Rogers–Szegő polynomials

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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.

References

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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