Mock modular forms and singular combinatorial series

Quantum modular forms and singular combinatorial series with repeated roots of unity

“…Remark 1.2. The conditions given in [12] do not require that gcd(α j , β j ) = 1. Instead, they merely require that α j β j = 1 2 Z.…”

“…Here, Π j is a constant depending only on ζ n,N as defined in [12], and A 3 is the level three Appell function (see [3] or [22])…”

“…where u, v ∈ C. In [22], Zwegers showed that A 3 (u, v; τ ) can be completed using the nonholomorphic function R 3 in (2.4) to transform like a non-holomorphic Jacobi form. Using these functions, as in [12] we let B n (ζ n,N ; q) := q − 1 24 (B + n (ζ n,N ; q) + B − n (ζ n,N ; q)), where the function B − n is given explicitly in terms of sums of functions involving F − m,3 (see (2.6)) and R 3 (see (2.4)) in [12, equation (4.3)]. We have the following theorem, established by two of the authors in [12].…”

“…Completing the function R n,N . Here we review some preliminary results and functions from [12]. Recall that n and N are fixed integers satisfying 0 ≤ N ≤ n 2 , and n ≥ 2.…”

“…with A 3 as defined in (1.7), R 3 as defined in (2.4), and Π m (x, w) as defined explicitly in [12]. The corresponding completed function is:…”