  In this paper, we consider the Fourier coefficients of a special class of
meromorphic Jaocbi forms of negative index. Much recent work has been done on
such coefficients in the case of Jacobi forms of positive index, but almost
nothing is known for Jacobi forms of negative index. In this paper we show,
from two different perspectives, that their Fourier coefficients have a simple
decomposition in terms of partial theta functions. The first perspective uses
the language of Lie super algebras, and the second applies the theory of
elliptic functions. In particular, we find a new infinite family of rank-crank
type PDEs generalizing the famous example of Atkin and Garvan. We then describe
the modularity properties of these coefficients, showing that they are "mixed
partial theta functions", along the way determining a new class of quantum
modular partial theta functions which is of independent interest.
