  We establish precise Zhu reduction formulas for Jacobi $n$-point functions
which show the absence of any possible poles arising in these formulas. We then
exploit this to produce results concerning the structure of strongly regular
vertex operator algebras, and also to motivate new differential operators
acting on Jacobi forms. Finally, we apply the reduction formulas to the Fermion
model in order to create polynomials of quasi-Jacobi forms which are Jacobi
forms.
