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 $k(τ)=r(τ)r^2(2τ)$ for the Ramanujan's cubic continued fraction, where $r(τ)$ is the Rogers-Ramanujan continued fraction, we use the methods of Lee and Park to study the modularity and arithmetic of the function $w(τ) = X(τ)X(3τ)$, which may be considered as an analogue of $k(τ)$ for the continued fraction $X(τ)$ of order six introduced by Vasuki, Bhaskar and Sharath. In particular, we show that $w(τ)$ can be written in terms of the normalized generator $u(τ)$ of the field of all modular functions on $Γ_0(18)$, and derive modular equations for $u(τ)$ of smaller prime levels. We also express $j(dτ)$ for $d\in\{1,2,3,6,9,18\}$ in terms of $u(τ)$, where $j$ is the modular $j$-invariant.