Ramanujan's continued fractions of order $10$ as modular functions

Abstract

We explore the modularity of the continued fractions $I(τ), J(τ), T_1(τ), T_2(τ)$ and $U(τ)=I(τ)/J(τ)$ of order $10$, where $I(τ)$ and $J(τ)$ are introduced by Rajkhowa and Saikia, which are special cases of certain identities of Ramanujan. In particular, we show that these fractions can be expressed in terms of an $η$-quotient $g(τ)$ that generates the field of all modular functions on the congruence subgroup $Γ_0(10)$. Consequently, we prove that modular equations for $g(τ)$ and $U(τ)$ exist at any level and derive these equations of prime levels $p\leq 11$. We also show that the continued fractions of order $10$ can be explicitly evaluated using a singular value of $g(τ)$, which under certain conditions, generates the Hilbert class field of an imaginary quadratic field. We employ the methods of Lee and Park to establish our results.