Modularity and the distinct rank function

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 ̂ R(ω,q) := R(ω,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 ̂ G(ω,q) :=G(ω,q−1) appears naturally in the theory of modular forms. Here we answer this question of Monks and Ono, and show that ̂ G(ω,q), when combined with ̂ G(ω−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 ̂ G(ω,q) may be deduced from this as a special case.