Rogers–Szegő 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 ) = ∑ 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[1]

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


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