Antiquantum $q$-series identities and mock theta functions
Amanda Folsom, David Metacarpa
TL;DR
This work broadens the landscape of antiquantum $q$-series by proving antiquantum identities for all Ramanujan–Watson third order mock theta functions. It fuses Lovejoy’s quantum $q$-series framework with the authors’ prior results and the modular theory of eta-quotients to produce boundary-root evaluations that match saturated interior series via dense root-of-unity sets. A key methodological advance is a general antiquantum identity (Theorem bid) combined with Watson relations and eta-quotient vanishing to extend results to all seven third-order mock theta functions, including explicit truncation factors. The results deepen the connection between mock theta functions, quantum modular forms, and modular objects, and provide concrete boundary-value identities that sharpen our understanding of convergence and truncation phenomena at roots of unity.
Abstract
Ramanujan's original definition of mock theta functions from 1920 involves their asymptotic behaviors at roots of unity on the boundary of the disk of convergence $|q|<1$. More recently this topic has been related by several authors, including the first author with Ono and Rhoades in 2013, to quantum modular forms, first defined in 2010 by Zagier. In 2021, Lovejoy defined and studied related quantum $q$-series identities, which do not hold as equalities between power series inside the disk $|q|<1$ but which do hold on dense subsets of roots of unity on the boundary. Inspired by this, in our prior joint work from 2024 we further studied quantum $q$-series identities as related to mock theta functions and quantum modular forms; we also defined and studied antiquantum $q$-series identities, between series which are equal inside the disk $|q|<1$ but which hold at dense sets of roots of unity on the boundary for which one of the series diverges and is unnaturally truncated. Here, building from our previous work, we establish antiquantum $q$-series identities for all of Ramanujan's third order mock theta functions. We deduce these results in part by establishing and applying more general identities which are also of independent interest, and by using the theory of modular eta-quotients.