On the Fourier coefficients of negative index meromorphic Jacobi forms

TL;DR

This paper studies the Fourier coefficients of negative index meromorphic Jacobi forms, extending previous work and connecting to important PDEs, partial theta functions, and quantum modular forms, revealing the structure of these coefficients.

Contribution

It extends the analysis of Fourier coefficients to negative index meromorphic Jacobi forms, linking to PDEs and quantum modular forms, and generalizing prior results.

Findings

Derived new formulas for Fourier coefficients of negative index meromorphic Jacobi forms.

Connected Fourier coefficients to PDEs like the rank-crank PDE of Atkin and Garvan.

Explored relations to partial theta functions and quantum modular forms.

Abstract

In this paper, we consider the Fourier coefficients of meromorphic Jacobi forms of negative index. This extends recent work of Creutzig and the first two authors for the special case of Kac-Wakimoto characters which occur naturally in Lie theory, and yields, as easy corollaries, many important PDEs arising in combinatorics such as the famous rank-crank PDE of Atkin and Garvan. Moreover, we discuss the relation of our results to partial theta functions and quantum modular forms as introducted by Zagier, which together with previous work on positive index meromorphic Jacobi forms illuminates the general structure of the Fourier coefficients of meromorphic Jacobi forms.