Mock Jacobi forms in basic hypergeometric series

TL;DR

This paper demonstrates that certain $q$-series, including mock theta functions, can be expressed as sums of theta quotients and mock Jacobi forms, revealing their modular properties and applications to partition theory.

Contribution

It shows that some $q$-series are linear sums of mock Jacobi forms and theta quotients, establishing their modularity and linking to partition statistics.

Findings

Mock theta functions are sums of theta quotients and mock Jacobi forms.

Linear sums of $q$-series can be weakly holomorphic modular forms of weight 1/2.

Relation between partition rank and crank derived from mock Jacobi forms.

Abstract

We show that some $q$-series such as universal mock theta functions are linear sums of theta quotients and mock Jacobi forms of weight 1/2, which become holomorphic parts of real analytic modular forms when they are restricted to torsion points and multiplied by suitable powers of $q$. And we prove that certain linear sums of $q$-series are weakly holomorphic modular forms of weight 1/2 due to annihilation of mock Jacobi forms or completion by mock Jacobi forms. As an application, we obtain a relation between the rank and crank of a partition.