Theta function identities
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.
scientific article; zbMATH DE number 4160842
scientific article