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

Abstract

On page 206 in his lost notebook, Ramanujan recorded the following enigmatic identity for his theta function $\phi (q)$: \begin{equation*} φ(e^{-7π\sqrt{7}}) = 7^{-3/4}φ(e^{-π\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 $\phi (e^{-n\pi }),$ for $n=7,21,35,$ and $49$.

20 pages, revised version, Section 3 is corrected

Cited by in corpus (2)

• Explicit values for Ramanujan's theta function $\phi (q)$
• Cubic and quintic analogues of Ramanujan's septic theta function identity