Discrete Normal distribution and its relationship with Jacobi Theta functions

We introduce new, natural parameters in a formula defining a family of discrete Normal distributions. One of the parameters is closely related to the expectation and the other to the variance of that family. We show that under such a parametrization, uniformly for all sufficiently large variances and all expectations, discrete Normal distributions and their first two moments are given by very simple formulae. We indicate the relation between our results and Jacobi Theta functions and Jacobi summation formulae.