---
tag: 7H7SACRYEF
kind: concept
name: Rogers polynomials
msc: [33D45]
sources:
  - url: https://en.wikipedia.org/wiki/Rogers_polynomials
same_as:
  wikidata: Q7359351
  wikipedia: 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|5TRFA6F79B]]: 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|M24H4GJPPW]].
They are special cases of the [[Askey–Wilson polynomials|RSNZNG0WA1]] and contain the [[continuous q-Hermite polynomials|XFKVZJ55Q5]] as the case $\beta = 0$.

## Explanations

### L1 human

The Rogers polynomials are defined by one generating function, a ratio of four infinite products.
Changing the parameter $\beta$ moves between several known families.
At $\beta = q$ they are the Chebyshev polynomials of the second kind, and at $\beta = 0$ they are the continuous $q$-Hermite polynomials up to a factor.
For example, $C_1(x; \beta \mid q) = \frac{2(1 - \beta)}{1 - q}\, x$.