Modularity of certain products of the Rogers-Ramanujan continued fraction

TL;DR

This paper investigates the modularity properties of products involving the Rogers-Ramanujan continued fraction and establishes conditions under which these products generate the field of modular functions for a specific subgroup, also exploring their arithmetic properties.

Contribution

It characterizes when products of the Rogers-Ramanujan continued fraction generate modular function fields and studies their arithmetic properties using eta-quotients.

Findings

Finitely many such functions generate the modular function field for (10).

Established arithmetic properties of the function l().

Applied eta-quotients techniques to prove modularity results.

Abstract

We study the modularity of the functions of the form $r(\tau)^ar(2\tau)^b$, where $a$ and $b$ are integers with $(a,b)\neq (0,0)$ and $r(\tau)$ is the Rogers-Ramanujan continued fraction, which may be considered as companions to the Ramanujan's function $k(\tau)=r(\tau)r(2\tau)^2$. 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 $\Gamma_1(10)$. Furthermore, we establish certain arithmetic properties of the function $l(\tau)=r(2\tau)/r(\tau)^2$, which can be used to evaluate these products. We employ the methods of Lee and Park, and some properties of $\eta$-quotients and generalized $\eta$-quotients to prove our results.