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