Theta function identities (Q919388)

All but one of the identities in the 21 chapters of Ramanujan's second Notebook had been proved by 1986. This paper proves the remaining identity, which involves a ratio of Dedekind eta functions \(\eta^ 2(z/p)/\eta^ 2(z)\) with \(p=13\). The author's method is quite general and constructs, for each odd integer \(p>1\), a class of modular functions on a subgroup \(\Gamma^ 0(p)\) of the modular group with no poles in the upper half plane or at the cusp 0. This yields a number of interesting identities involving the ratio \(\eta^ 2(z/p)/\eta^ 2(z)\) for any odd \(p>1\). These include the case with \(p=13\) mentioned above and several other special cases with \(p=5,7,9\), and 17 stated by Ramanujan.