Appell Functions for General Lattices

We study Appell functions associated to an arbitrary positive definite lattice $\Lambda$ and a choice of $M\leq {\rm dim}(\Lambda)$ linearly independent vectors $d_r\in \Lambda$, $r=1,\dots,M$. These functions are instances of multi-variable quasi-elliptic functions, and specific examples have appeared at various places in mathematics and theoretical physics. For example, if $\Lambda$ is chosen to be one-dimensional, these functions reduce to the classical Appell function, which is a prominent example in the theory of mock modular forms. The Appell functions introduced here are examples of depth $M$ mock modular forms. We derive a structural formula for their modular completion. Motivated by partition functions in theoretical physics, we discuss the case where $\Lambda$ is the $A_N$ root lattice in detail.