Rogers polynomials

Summary

The Rogers polynomials $C_n(x; \beta \mid q)$, also called the continuous $q$-ultraspherical polynomials, are a family of orthogonal polynomials with a parameter $\beta$ and a base $q$. They are the $q$-analogue of the Gegenbauer polynomials: with $\beta = q^\lambda$ they tend to $C_n^{(\lambda)}(x)$ as $q \to 1$. L. J. Rogers used them in his work that led to the Rogers–Ramanujan identities. They are special cases of the Askey–Wilson polynomials and contain the continuous q-Hermite polynomials as the case $\beta = 0$.