Modularity and the distinct rank function

Abstract

If R(ω,q) denotes Dyson’s partition rank generating function, due to work of Bringmann and Ono, it is known that for roots of unity ω≠1, R(ω,q) is the “holomorphic part” of a harmonic weak Maass form. Dating back to Ramanujan, it is also known that \(\widehat{R}(\omega,q):=R(\omega,q^{-1})\) is given by Eichler integrals and modular forms. In analogy to these results, more recently Monks and Ono have shown that modular forms arise in a natural way from G(ω,q), the generating function for ranks of partitions into distinct parts. Moreover, Monks and Ono pose the following problem: determine whether the function \(\widehat{G}(\omega,q):=G(\omega,q^{-1})\) appears naturally in the theory of modular forms. Here we answer this question of Monks and Ono, and show that \(\widehat{G}(\omega,q)\) , when combined with \(\widehat{G}(\omega^{-1},q)\) and a twisted third-order mock theta of Ramanujan, form a weight 1 modular form. We provide a more general result on the modularity of certain expressions involving basic hypergeometric series and then show that our result on \(\widehat {G}(\omega,q)\) may be deduced from this as a special case.