Rogers polynomials should not be confused with Rogers–Szegő polynomials.
In mathematics, the Rogers polynomials, also called Rogers–Askey–Ismail polynomials and continuous q-ultraspherical polynomials, are a family of orthogonal polynomials introduced by in the course of his work on the Rogers–Ramanujan identities. They are q-analogs of ultraspherical polynomials, and are the Macdonald polynomials for the special case of the A1 affine root system .
and discuss the properties of Rogers polynomials in detail.
The Rogers polynomials can be defined in terms of the q-Pochhammer symbol and the basic hypergeometric series by
Cn(x;\beta|q)=
| (\beta;q)n | |
| (q;q)n |
ein\theta{}2\phi
| -n | |
| 1(q |
,\beta;\beta-1q1-n;q,q\beta-1e-2i\theta)