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  "body": "\n## Summary\n\nThe 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 $q$.\nThey are the $q$-analogue of the [[Gegenbauer polynomials|5TRFA6F79B]]: with $\\beta = q^\\lambda$ they tend to $C_n^{(\\lambda)}(x)$ as $q \\to 1$.\nL. J. Rogers used them in his work that led to the [[Rogers–Ramanujan identities|M24H4GJPPW]].\nThey are special cases of the [[Askey–Wilson polynomials|RSNZNG0WA1]] and contain the [[continuous q-Hermite polynomials|XFKVZJ55Q5]] as the case $\\beta = 0$.\n\n## Explanations\n\n### L1 human\n\nThe Rogers polynomials are defined by one generating function, a ratio of four infinite products.\nChanging the parameter $\\beta$ moves between several known families.\nAt $\\beta = q$ they are the Chebyshev polynomials of the second kind, and at $\\beta = 0$ they are the continuous $q$-Hermite polynomials up to a factor.\nFor example, $C_1(x; \\beta \\mid q) = \\frac{2(1 - \\beta)}{1 - q}\\, x$.\n",
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