Identities for the Rogers-Ramanujan continued fraction
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We prove some new modular identities for the Rogers-\linebreak Ramanujan continued fraction. For example, if $R(q)$ denotes the Rogers-Ramanujan continued fraction, then \begin{align*}&R(q)R(q^4)=\dfrac{R(q^5)+R(q^{20})-R(q^5)R(q^{20})}{1+R(q^{5})+R(q^{20})},\\ &\dfrac{1}{R(q^{2})R(q^{3})}+R(q^{2})R(q^{3})= 1+\dfrac{R(q)}{R(q^{6})}+\dfrac{R(q^{6})}{R(q)}, \end{align*} and \begin{align*}&R(q^2)\\ =&\ \dfrac{R(q)R(q^3)}{R(q^6)}\cdot\dfrac{R(q) R^2(q^3) R(q^6)+2 R(q^6) R(q^{12})+ R(q) R(q^3) R^2(q^{12})}{R(q^3) R(q^6)+2 R(q) R^2(q^3) R(q^{12})+ R^2(q^{12})}.\end{align*} In the process, we also find some new relations for the Rogers-Ramanujan functions by using dissections of theta functions and the quintuple product identity.
Abstract
We prove some new modular identities for the Rogers-\linebreak Ramanujan continued fraction. For example, if $R(q)$ denotes the Rogers-Ramanujan continued fraction, then \begin{align*}&R(q)R(q^4)=\dfrac{R(q^5)+R(q^{20})-R(q^5)R(q^{20})}{1+R(q^{5})+R(q^{20})},\\ &\dfrac{1}{R(q^{2})R(q^{3})}+R(q^{2})R(q^{3})= 1+\dfrac{R(q)}{R(q^{6})}+\dfrac{R(q^{6})}{R(q)}, \end{align*} and \begin{align*}&R(q^2)\\ =&\ \dfrac{R(q)R(q^3)}{R(q^6)}\cdot\dfrac{R(q) R^2(q^3) R(q^6)+2 R(q^6) R(q^{12})+ R(q) R(q^3) R^2(q^{12})}{R(q^3) R(q^6)+2 R(q) R^2(q^3) R(q^{12})+ R^2(q^{12})}.\end{align*} In the process, we also find some new relations for the Rogers-Ramanujan functions by using dissections of theta functions and the quintuple product identity.