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

[math]\displaystyle{ 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)_n is the descending q-Pochhammer symbol.

Furthermore, the [math]\displaystyle{ h_n(x;q) }[/math] satisfy (for [math]\displaystyle{ n \ge 1 }[/math]) the recurrence relation

[math]\displaystyle{ h_{n+1}(x;q) = (1+x)h_n(x;q) + x(q^n-1)h_{n-1}(x;q) }[/math]

with [math]\displaystyle{ h_0(x;q)=1 }[/math] and [math]\displaystyle{ h_1(x;q)=1+x }[/math].

References

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