Extending a catalog of mock and quantum modular forms to an infinite class

TL;DR

This paper introduces an infinite class of functions that generalize previous mock and quantum modular forms, expanding the catalog and demonstrating their dual properties.

Contribution

It extends the existing catalog by defining an infinite class of functions that are both mock modular and quantum modular forms, generalizing a specific subset.

Findings

The new functions form an infinite class extending previous examples.

These functions exhibit both mock modular and quantum modular properties.

The work broadens understanding of the structure and classification of modular forms.

Abstract

Utilizing a classification due to Lemke Oliver of eta-quotients which are also theta functions (here called eta-theta functions), Folsom, Garthwaite, Kang, Treneer, and the fourth author constructed a catalog of mock modular forms $V_{mn}$ having weight $3/2$ eta-theta function shadows and showed that these mock modular forms when viewed on certain sets of rationals, transform as quantum modular forms under the action of explicit subgroups. In this paper, we introduce an infinite class of functions that generalizes one row of the catalog, namely the $V_{m1}$, and show that the functions in this infinite class are both mock modular and quantum modular forms.