Explicit evaluations of a level 13 analogue of the Rogers-Ramanujan continued fraction

Abstract: In his first and lost notebooks, Ramanujan recorded some exact numerical values for the Rogers-Ramanujan continued fraction \(q^{1/5}\prod_{j=1}^{\infty}\left(1-q^{j}\right)^{\left(\frac{j}{5}\right)},\) for specific values of $q$. Here, $\left(\frac{j}{p}\right)$ is the Legendre symbol. In this work, we give explicit evaluations of an analogue of the Rogers-Ramanujan continued fraction defined by \(q\prod_{j=1}^{\infty}\left(1-q^{j}\right)^{\left(\frac{j}{13}\right)}.\)