Applications of a new P-Q modular equation of degree two

Abstract

At scattered places in his first notebook, Ramanujan recorded the values for 107 class invariants or irreducible monic polynomials satisfied by them. On pages 294-299 in his second notebook, he gave a table of values for 77 class invariants $G_n$ and $g_n$ in his second notebook. Traditionally, $G_n$ is determined for odd values of $n$ and $g_n$ for even values of $n$. On pages 338 and 339 in his first notebook, Ramanujan defined the remarkable product of theta-functions $a_{m, n}$. Also, he recorded eighteen explicit values depending on two parameters, namely, $m$, and $n$, where these are odd integers. In this paper, we initiate to study explicit evaluations of $G_n$ for even values of $n$. We establish a new general formula for the explicit evaluations of $G_n$ involving class invariant $g_n$. For this purpose, we derive a new P-Q modular equation of degree two. Further application of this modular equation, we establish a new formula to explicit evaluation of $a_{m, 2}$. Also, we compute several explicit values of class invariant $g_{n}$ and singular moduli $\alpha_n$.