We study the modularity of the functions of the form \(r(\tau )^ar(2\tau )^b\) , where a and b are integers with \((a,b)\ne (0,0)\) 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.
