The three missing terms in Ramanujan's septic theta function identity

TL;DR

This paper completes Ramanujan's septic theta function identity by identifying the missing terms, and computes related class invariants and special values of the theta function for specific arguments.

Contribution

It provides the missing terms in Ramanujan's septic theta function identity and calculates new class invariants and special theta function values.

Findings

Identified the three missing terms in Ramanujan's identity.

Calculated the class invariant G_{343}.

Derived special values of (e^{-n\u03c0}) for n=7, 21, 35, 49.

Abstract

On page 206 in his lost notebook, Ramanujan recorded the following enigmatic identity for his theta function $\varphi(q)$: \begin{equation*} \varphi(e^{-7\pi\sqrt{7}}) = 7^{-3/4}\varphi(e^{-\pi\sqrt{7}})\big\{1 + (\quad)^{2/7} + (\quad)^{2/7} + (\quad)^{2/7}\big\}. \end{equation*} We give the three missing terms. In addition, we calculate the class invariant $G_{343}$ and further special values of $\varphi(e^{-n\pi}),$ for $n = 7, 21, 35,$ and $49$.