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Identities for the Rogers-Ramanujan Continued Fraction (2410.17110v1)

Published 22 Oct 2024 in math.NT

Abstract: We prove some new modular identities for the Rogers\textendash Ramanujan continued fraction. For example, if $R(q)$ denotes the Rogers\textendash Ramanujan continued fraction, then \begin{align*}&R(q)R(q4)=\dfrac{R(q5)+R(q{20})-R(q5)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(q2)=\dfrac{R(q)R(q3)}{R(q6)}\cdot\dfrac{R(q) R2(q3) R(q6)+2 R(q6) R(q{12})+ R(q) R(q3) R2(q{12})}{R(q3) R(q6)+2 R(q) R2(q3) R(q{12})+ R2(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.

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