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 φ(q):

φ(e^{−7π√7}) = 7^{−3/4}φ(e^{−π√7}){1 + ( )^{2/7} + ( )^{2/7} + ( )^{2/7}}.

We give the three missing terms. In addition, we calculate the class invariant G_{343} and further special values of φ(e^{−nπ}), for n = 7, 21, 35, and 49.