  The topological vertex is a universal series which can be regarded as an
object in combinatorics, representation theory, geometry, or physics. It
encodes the combinatorics of 3D partitions, the action of vertex operators on
Fock space, the Donaldson-Thomas theory of toric Calabi-Yau threefolds, or the
open string partition function of $\mathbb{C}^3$.
  We prove several identities in which a sum over terms involving the
topological vertex is expressed as a closed formula, often a product of simple
terms, closely related to Fourier expansions of Jacobi forms. We use purely
combinatorial and representation theoretic methods to prove our formulas, but
we discuss applications to the Donaldson-Thomas invariants of elliptically
fibered Calabi-Yau threefolds at the end of the paper.
