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  "body": "\n## Definition\n\nLet $q, \\beta \\in \\C$ with $\\abs{q} \u003c 1$ and $\\abs{\\beta} \u003c 1$, and let $x = \\cos\\theta$.\nThe *Rogers polynomials* $C_n(x; \\beta \\mid q)$, $n \\ge 0$, are the coefficients of the [[power series|0MZB06ST2B]] in $t$\n$$\\sum_{n=0}^{\\infty} C_n(x; \\beta \\mid q)\\, t^n = \\frac{(\\beta t e^{i\\theta}; q)_\\infty\\, (\\beta t e^{-i\\theta}; q)_\\infty}{(t e^{i\\theta}; q)_\\infty\\, (t e^{-i\\theta}; q)_\\infty}, \\qquad \\abs{t} \u003c 1,$$\nwhere $(\\cdot\\,; q)_\\infty$ is the infinite [[q-Pochhammer symbol|Y1X8ZZQ7VM]].\n\n## Remarks\n\nExpanding each factor by the $q$-binomial theorem gives the explicit form\n$$C_n(x; \\beta \\mid q) = \\sum_{k=0}^{n} \\frac{(\\beta; q)_k\\, (\\beta; q)_{n-k}}{(q; q)_k\\, (q; q)_{n-k}}\\, e^{i(n - 2k)\\theta}.$$\nThe terms for $k$ and $n - k$ pair to $2\\cos((n - 2k)\\theta)$ times the same coefficient.\nSo $C_n$ is a polynomial in $x$ of degree $n$, with leading coefficient $2^n \\frac{(\\beta; q)_n}{(q; q)_n} \\ne 0$.\n\nExample: $\\beta = q$.\nThen $\\frac{(q; q)_k (q; q)_{n-k}}{(q; q)_k (q; q)_{n-k}} = 1$, 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]].\n\nNon-example: $\\beta = 1$ fails the condition $\\abs{\\beta} \u003c 1$.\nThere 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 $n$.\n\nFor $\\beta = 0$, $C_n(x; 0 \\mid q) = \\frac{H_n(x \\mid q)}{(q; q)_n}$, with the [[continuous q-Hermite polynomials|XFKVZJ55Q5]].\nFor $0 \u003c q \u003c 1$ and $-1 \u003c \\beta \u003c 1$, 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 $d\\theta$, $x = \\cos\\theta$.\nAs $q \\to 1$, $C_n(x; q^\\lambda \\mid q) \\to C_n^{(\\lambda)}(x)$, the [[Gegenbauer polynomial|5TRFA6F79B]].\n",
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    "name": "Rogers polynomials",
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  "name": "Rogers polynomials",
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