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