  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.
