Modularity of certain products of the Rogers-Ramanujan continued fraction

Abstract

We study the modularity of the functions of the form $r(τ)^ar(2τ)^b$, where $a$ and $b$ are integers with $(a,b)\neq (0,0)$ and $r(τ)$ is the Rogers-Ramanujan continued fraction, which may be considered as companions to the Ramanujan's function $k(τ)=r(τ)r(2τ)^2$. In particular, we show that under some condition on $a$ and $b$, there are finitely many such functions generating the field of all modular functions on the congruence subgroup $Γ_1(10)$. Furthermore, we establish certain arithmetic properties of the function $l(τ)=r(2τ)/r(τ)^2$, which can be used to evaluate these products. We employ the methods of Lee and Park, and some properties of $η$-quotients and generalized $η$-quotients to prove our results.