Analogue of Ramanujan's function $k(τ)$ for the continued fraction $X(τ)$ of order six

Abstract: 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 $\Gamma_0(18)$, 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.