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  "body": "\n## Definition\n\nLet $\\tau$ be in the [[upper half-plane|X6XGGH5FPW]], let $q = e^{2\\pi i \\tau}$, and let $q^{1/5} = e^{2\\pi i \\tau / 5}$.\nThe *Rogers–Ramanujan continued fraction* is\n$$R(\\tau) = \\cfrac{q^{1/5}}{1 + \\cfrac{q}{1 + \\cfrac{q^2}{1 + \\cfrac{q^3}{1 + \\cdots}}}},$$\nthe limit of the finite [[continued fractions|04P2RJ4WPS]] obtained by stopping after the term $q^n$, as $n \\to \\infty$.\n\n## Remarks\n\nSince $\\abs{q} \u003c 1$ the limit exists, and Rogers and Ramanujan showed\n$$R(\\tau) = q^{1/5} \\prod_{n=1}^{\\infty} \\frac{(1 - q^{5n-1})(1 - q^{5n-4})}{(1 - q^{5n-2})(1 - q^{5n-3})} = q^{1/5}\\, \\frac{H(q)}{G(q)},$$\nwith $G(q) = \\sum_{n \\ge 0} \\frac{q^{n^2}}{(q; q)_n}$ and $H(q) = \\sum_{n \\ge 0} \\frac{q^{n^2 + n}}{(q; q)_n}$, the two sides of the [[Rogers–Ramanujan identities|M24H4GJPPW]].\n\nExample: $\\tau = i$, so $q = e^{-2\\pi} \\approx 0.001867$.\nThen $R(i) = \\sqrt{\\frac{5 + \\sqrt5}{2}} - \\frac{1 + \\sqrt5}{2} \\approx 0.284079$.\nA numerical check: $q^{1/5} = e^{-2\\pi/5} \\approx 0.284609$, and the continued fraction $1 + q/(1 + q^2/(1 + \\cdots))$ is about $1.001867$, whose quotient is about $0.284079$.\n\nNon-example: with numerators $1, 1, 1, \\dots$ instead of $q, q^2, q^3, \\dots$, the fraction $\\cfrac{1}{1 + \\cfrac{1}{1 + \\cdots}}$ is $\\frac{\\sqrt5 - 1}{2}$, the inverse of the golden ratio, and it does not depend on $q$; it is not $R$.\n\nAnother common notation is $R(q)$ in place of $R(\\tau)$.\n",
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