Quantum modularity of partial theta series with periodic coefficients

Abstract

We explicitly prove the quantum modularity of partial theta series with even or odd periodic coefficients. As an application, we show that the Kontsevich-Zagier series ${\mathcal{F}}_t(q)$ which matches (at a root of unity) the colored Jones polynomial for the family of torus knots $T(3,2^t)$, $t\ge 2$, is a weight $3/2$ quantum modular form. This generalizes Zagier's result on the quantum modularity for the "strange" series $F(q)$.

16 pages, to appear in Forum Mathematicum

References in corpus (2)

• q-series and L-functions related to half-derivatives of the Andrews--Gordon identity
• Generalized Fishburn numbers and torus knots

Cited by in corpus (2)

• Generalized Fishburn numbers and torus knots
• Asymptotics and sign patterns for coefficients in expansions of Habiro elements