Misplaced Pages

Rogers–Szegő polynomials

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Not to be confused with Rogers polynomials.
This article includes inline citations, but they are not properly formatted. Please improve this article by correcting them. (May 2024) (Learn how and when to remove this message)

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 ) = ∑ k = 0 n ( q ; q ) n ( q ; q ) k ( q ; q ) n − k x k {\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 h n ( x ; q ) {\displaystyle h_{n}(x;q)} satisfy (for n ≥ 1 {\displaystyle n\geq 1} ) the recurrence relation

h n + 1 ( x ; q ) = ( 1 + x ) h n ( x ; q ) + x ( q n − 1 ) h n − 1 ( x ; q ) {\displaystyle 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 {\displaystyle h_{0}(x;q)=1} and h 1 ( x ; q ) = 1 + x {\displaystyle h_{1}(x;q)=1+x} .

References

  1. Vinroot, C. Ryan (12 July 2012). "An enumeration of flags in finite vector spaces". The Electronic Journal of Combinatorics. 19 (3). doi:10.37236/2481.
  • Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press, 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


Stub icon

This polynomial-related article is a stub. You can help Misplaced Pages by expanding it.

Categories: