The Rogers-Ramanujan continued fraction and its level 13 analogue

Abstract: One of the properties of the Rogers-Ramanujan continued fraction is its representation as an infinite product given by \(\scriptr(q) = q^{1/5}\prod_{j=1}^\infty (1-q^j)^{\left(\frac{j}{5}\right)}\) where $\left(\frac{j}{p}\right)$ is the Legendre symbol. In this work we study the level $13$ function \(R(q) = q\prod_{j=1}^\infty (1-q^j)^{\left(\frac{j}{13}\right)}\) and establish many properties analogous to those for the fifth power of the Rogers-Ramanujan continued fraction. Many of the properties extend to other levels $\ell$ for which $\ell-1$ divides $24$, and a brief account of these results is included.