``Strange'' combinatorial quantum modular forms (Q311480)

The subject of quantum modular forms is relatively young. Hence it has been of recent interest to further explore the theory and to find explicit examples. In the present paper, the authors define a natural two-variable ``strange'' function NEWLINE\[NEWLINEF(w; q):=\sum_{n=0}^{\infty} w^{n+1} (wq; q)_n,NEWLINE\]NEWLINE where \((a; q)_n:=\prod_{j=0}^{n-1} (1-a q^j)\) for \(n \in {\mathbb N}\) and \((a; q)_0:=1\). In particular, \(F(1; q)=F(q)\) is Kontsevich's strange function, a quantum modular form. They show that when \(q\) is a primitive \(k\)-th root of unity, and \(w\) is any complex number in certain subsets in \({\mathbb C}\), the two-variable unimodal rank generating function \(U(w; q)\) is equal to their two-variable strange function \(F(w; q^{-1})\). As a consequence, they show that radial limits of the difference of the mock modular partition rank and modular crank generating functions can be expressed as special values of \(F(w; q)\). Moreover, the authors show that \(F(w; q)\) can be used to define infinite families of quantum modular forms.