Appell functions for general lattices

We study Appell functions associated with an arbitrary positive definite lattice Λ Λ and a choice of M≤ dim(Λ ) M ≤ dim ( Λ ) linearly independent vectors dᵣ∈ Λ d r ∈ Λ , r=1, ,M r = 1 , ⋯ , 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 Λ Λ 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 an explicit structural formula for their modular completion. Motivated by partition functions in theoretical physics, we discuss the case where Λ Λ is the AN A N root lattice in detail.