Quantum modular forms

example

• unimodular generating function

$$ U(w;q)=\sum_{n=0}^{\infty}(wq;q)_{n}(w^{-1}q;q)_{n}q^{n+1} $$

• Dyson's rank generating function

$$R(w;q)=\sum_{n=0}^\infty \frac{q^{n^2}}{(wq;q)_n(w^{-1}q;q)_n}$$

• Andrews-Garvan crank modular form

$$C(w;q)=\frac{(q)_{\infty}}{(wq;q)_{\infty}(w^{-1}q;q)_{\infty}}$$

• limit formula $\zeta_b=e^{2\pi i/b}$, $1\le a <b$, for every root of unity $\zeta$, there exists an integer $c$ such that

$$ \lim_{q\to \zeta} R(\zeta_{b}^{a};q)-\zeta_{b^2}^{c} C(\zeta_{b}^{a};q)=-(1-\zeta_{b}^{a})(1-\zeta_{b}^{-a})U(\zeta_{b}^{a};\zeta) $$

special case

• If $b=2$ and $a=1$, then $\zeta_{b}^{a}=-1$
• $U(-1;\zeta)$ becomes a finite sum if $\zeta$ is a root of unity

$$ U(-1;\zeta)=\sum_{n=0}^{k-1} (1+\zeta)^2(1+\zeta^2)^2\cdots (1+\zeta^n)^2\zeta^{n+1} $$

• $R(-1;q)=f(q)$ and $C(-1;q)=b(q)$ in 3rd order mock theta functions

related items

• quantum dilogarithm

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxWmFPWkZTMVdBeDA/edit