Analogue of Ramanujan’s function \(k(\tau )\) for the continued fraction \(X(\tau )\) of order six

Motivated by the recent work of Park on the analogue of the Ramanujan’s function \(k(\tau )=r(\tau )r^2(2\tau )\) $k(\tau )=r(\tau )r^2(2\tau )$ for the Ramanujan’s cubic continued fraction, where \(r(\tau )\) $r(\tau )$ is the Rogers–Ramanujan continued fraction, we use the methods of Lee and Park to study the modularity and arithmetic of the function \(w(\tau ) = X(\tau )X(3\tau )\) $w(\tau )=X(\tau )X(3\tau )$ , which may be considered as an analogue of \(k(\tau )\) $k(\tau )$ for the continued fraction \(X(\tau )\) $X(\tau )$ of order six introduced by Vasuki, Bhaskar and Sharath. In particular, we show that \(w(\tau )\) $w(\tau )$ can be written in terms of the normalized generator \(u(\tau )\) $u(\tau )$ of the field of all modular functions on \(\Gamma _0(18)\) ${\mathrm{\Gamma }}_0(18)$ , and derive modular equations for \(u(\tau )\) $u(\tau )$ of smaller prime levels. We also express \(j(d\tau )\) $j(d\tau )$ for \(d\in \{1,2,3,6,9,18\}\) $d\in \{1,2,3,6,9,18\}$ in terms of \(u(\tau )\) $u(\tau )$ , where j is the modular j-invariant.