Applications of the icosahedral equation for the Rogers-Ramanujan continued fraction

Abstract

Let $R(q)$ denote the Rogers-Ramanujan continued fraction for $|q| < 1$. By applying the RootApproximant command in the Wolfram language to expressions involving the theta function $f(-q) := (q;q)_{\infty}$ given in modular relations due to Yi, this provides a systematic way of obtaining experimentally discovered evaluations for $R\big(e^{-\pi\sqrt{r}}\big)$, for $r \in \mathbb{Q}_{> 0}$. We succeed in applying this approach to obtain explicit closed forms, in terms of radicals over $\mathbb{Q}$, for the Rogers-Ramanujan continued fraction that have not previously been discovered or proved. We prove our closed forms using the icosahedral equation for $R$ together with closed forms for and modular relations associated with Ramanujan's $G$- and $g$-functions. An especially remarkable closed form that we introduce and prove is for $R\big( e^{-\pi \sqrt{48/5} } \big)$, in view of the computational difficulties surrounding the application of an order-25 modular relation in the evaluation of $G_{48/5}$.