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 $p\le 11$. 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.

21 pages; made minor changes as suggested by referees