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

Motivated by the recent work of Park on the analogue of the Ramanujan’s function k (τ) = r (τ) r 2 (2 τ) 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 (3 τ) , 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 (d τ) for d ∈ { 1 , 2 , 3 , 6 , 9 , 18 } in terms of u (τ) , where j is the modular j-invariant. [ABSTRACT FROM AUTHOR]