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

Abstract

We explore the modularity of the continued fractions $I(\tau), J(\tau), T_1(\tau), T_2(\tau)$ and $U(\tau)=I(\tau)/J(\tau)$ of order $10$, where $I(\tau)$ and $J(\tau)$ 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 $\eta$-quotient $g(\tau)$ that generates the field of all modular functions on the congruence subgroup $\Gamma_0(10)$. Consequently, we prove that modular equations for $g(\tau)$ and $U(\tau)$ 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 $g(\tau)$, 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.