---
tag: 6CW61T5BB2
kind: definition
name: Rogers polynomials
msc: [33D45]
defines: 7H7SACRYEF
uses: [Y1X8ZZQ7VM, 0MZB06ST2B]
sources:
  - url: https://en.wikipedia.org/wiki/Rogers_polynomials
---

## Definition

Let $q, \beta \in \C$ with $\abs{q} < 1$ and $\abs{\beta} < 1$, and let $x = \cos\theta$.
The *Rogers polynomials* $C_n(x; \beta \mid q)$, $n \ge 0$, are the coefficients of the [[power series|0MZB06ST2B]] in $t$
$$\sum_{n=0}^{\infty} C_n(x; \beta \mid q)\, t^n = \frac{(\beta t e^{i\theta}; q)_\infty\, (\beta t e^{-i\theta}; q)_\infty}{(t e^{i\theta}; q)_\infty\, (t e^{-i\theta}; q)_\infty}, \qquad \abs{t} < 1,$$
where $(\cdot\,; q)_\infty$ is the infinite [[q-Pochhammer symbol|Y1X8ZZQ7VM]].

## Remarks

Expanding each factor by the $q$-binomial theorem gives the explicit form
$$C_n(x; \beta \mid q) = \sum_{k=0}^{n} \frac{(\beta; q)_k\, (\beta; q)_{n-k}}{(q; q)_k\, (q; q)_{n-k}}\, e^{i(n - 2k)\theta}.$$
The terms for $k$ and $n - k$ pair to $2\cos((n - 2k)\theta)$ times the same coefficient.
So $C_n$ is a polynomial in $x$ of degree $n$, with leading coefficient $2^n \frac{(\beta; q)_n}{(q; q)_n} \ne 0$.

Example: $\beta = q$.
Then $\frac{(q; q)_k (q; q)_{n-k}}{(q; q)_k (q; q)_{n-k}} = 1$, so $C_n(x; q \mid q) = \sum_{k=0}^{n} e^{i(n-2k)\theta} = \frac{\sin((n+1)\theta)}{\sin\theta}$, the Chebyshev polynomial $U_n(x)$ of the [[second kind|7CATGEN6KH]].

Non-example: $\beta = 1$ fails the condition $\abs{\beta} < 1$.
There the right side of the generating function is $1$, so all coefficients of $t^n$ with $n \ge 1$ vanish, and there is no polynomial of degree $n$.

For $\beta = 0$, $C_n(x; 0 \mid q) = \frac{H_n(x \mid q)}{(q; q)_n}$, with the [[continuous q-Hermite polynomials|XFKVZJ55Q5]].
For $0 < q < 1$ and $-1 < \beta < 1$, the $C_n$ are orthogonal on $[-1, 1]$ for the weight $\frac{(e^{2i\theta}, e^{-2i\theta}; q)_\infty}{(\beta e^{2i\theta}, \beta e^{-2i\theta}; q)_\infty}$ with respect to $d\theta$, $x = \cos\theta$.
As $q \to 1$, $C_n(x; q^\lambda \mid q) \to C_n^{(\lambda)}(x)$, the [[Gegenbauer polynomial|5TRFA6F79B]].