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

Abstract We study the modularity of the function w(τ ) = X(τ )X(3τ ), where X(τ ) is the continued fraction of order six introduced by Vasuki, Bhaskar and Sharath (2010). The function w(τ ) is an analogue of Ramanujan’s function k(τ ) = r(τ )r(2τ ) 2 , where r(τ ) is the Rogers-Ramanujan continued fraction. We prove that w(τ ) is an η-quotient that generates the field of all modular functions on the congruence subgroup Γ 0 (18), and express X(τ ) and X(3τ ) in terms of w(τ ). We also show that there is a modular equation for w(τ ) of any level, and provide explicitly the modular equations of level p for primes p ≤ 13. We finally show that w( τ/3 ) generates the ray class field K(3) over an imaginary quadratic field K modulo 3 for some suitable τ ∈ K.