Modularity of certain products of the Rogers-Ramanujan continued fraction

In this paper, we study the modularity of the functions of the form r(τ)ᵃr(2τ)ᵇ, where a and b are integers with (a,b)≠ (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 Γ₁(10). Furthermore, we establish certain arithmetic properties of the function l(τ)=r(τ)⁻²r(2τ), which can be used to evaluate these functions. We employ the methods of Lee and Park, and some properties of η-quotients and generalized η-quotients to prove our results.