Modular forms arising from Q(n) and Dysonʼs rank

Let R(w;q) be Dysonʼs generating function for partition ranks. For roots of unity ζ≠1, it is known that R(ζ;q) and R(ζ;1/q) are given by harmonic Maass forms, Eichler integrals, and modular units. We show that modular forms arise from G(w;q), the generating function for ranks of partitions into distinct parts, in a similar way. If D(w;q):=(1+w)G(w;q)+(1−w)G(−w;q), then for roots of unity ζ≠±1 we show that q112⋅D(ζ;q)D(ζ−1;q) is a weight 1 modular form. Although G(ζ;1/q) is not well defined, we show that it gives rise to natural sequences of q-series whose limits involve infinite products (and modular forms when ζ=1). Our results follow from work of Fine on basic hypergeometric series.