Mock Modular Forms and Singular Combinatorial Series

upon specializing q = qτ := e 2πiτ , τ ∈ H the upper complex half-plane, where η(τ) := q1/24 ∏ n≥1(1 − qn) is Dedekind’s η-function, a weight 1/2 modular form. More recently, Bringmann and Ono [8] studied the generating function for partition ranks, where the rank of a partition, after Dyson, is defined to be the largest part of the partition minus the number of parts. For example, the rank of the partition 2 + 1 + 1 is 2 − 3 = −1. If N(m,n) := #{partitions of n with rank equal to m}, it is well known that the associated two-variable generating function satisfies