View source for Rogers–Szegő polynomials

{{distinguish|Rogers polynomials}}
In mathematics, the '''Rogers–Szegő polynomials''' are a family of  [[polynomials orthogonal on the unit circle]] introduced by {{harvs|txt|last=Szegő|authorlink=Gábor Szegő|year=1926}}, who was inspired by the continuous ''q''-Hermite polynomials studied by [[Leonard James Rogers]]. They are given by
:<math>h_n(x;q) = \sum_{k=0}^n\frac{(q;q)_n}{(q;q)_k(q;q)_{n-k}}x^k</math>
where (''q'';''q'')<sub>''n''</sub> is the descending [[q-Pochhammer symbol]].

Furthermore, the <math>h_n(x;q)</math> satisfy (for <math>n \ge 1</math>) the recurrence relation<ref name = vinroot>{{cite journal|title = An enumeration of flags in finite vector spaces| first = C. Ryan| last = Vinroot |journal = The Electronic Journal of Combinatorics |date = 12 July 2012 |volume=19 |issue=3 | url = http://www.combinatorics.org/ojs/index.php/eljc/article/download/v19i3p5/pdf}}</ref>
:<math> h_{n+1}(x;q) = (1+x)h_n(x;q) + x(q^n-1)h_{n-1}(x;q)</math>
with <math>h_0(x;q)=1</math> and <math>h_1(x;q)=1+x</math>.

==References==
{{Reflist}}
*{{Citation | last1=Gasper | first1=George | last2=Rahman | first2=Mizan | title=Basic hypergeometric series | publisher=[[Cambridge University Press]] | edition=2nd | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-83357-8 | doi=10.2277/0521833574 |mr=2128719 | year=2004 | volume=96}}
*{{Citation | last1=Szegő | first1=Gábor | title=Beitrag zur theorie der thetafunktionen | id=Reprinted in his collected papers | year=1926 | journal=Sitz Preuss. Akad. Wiss. Phys. Math. Ki. | volume=XIX | pages=242–252}}

{{DEFAULTSORT:Rogers-Szego polynomials}}
[[Category:Orthogonal polynomials]]