Identities for the Rogers-Ramanujan Continued Fraction

TL;DR

This paper derives new modular identities involving the Rogers-Ramanujan continued fraction, expanding the understanding of its algebraic and modular properties through novel relations and dissections.

Contribution

It introduces new modular identities for the Rogers-Ramanujan continued fraction, including relations involving products and ratios, using theta function dissections and the quintuple product identity.

Findings

New identities for R(q) involving products and ratios.

Relations connecting R(q) at different powers of q.

Enhanced understanding of Rogers-Ramanujan functions.

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(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.