Discretisations of higher order and the theorems of Faà di Bruno and DeMoivre-Laplace

We study discrete functions on equidistant and non-equidistant infinitesimal grids. We consider their difference quotients of higher order and give conditions for their near-equality to the corresponding derivatives. Important tools are nonstandard notions of regularity of higher order, and the formula of Faà di Bruno for higher order derivatives and a discrete version of it. As an application of such transitions from the discrete to the continuous we extend the DeMoivre-Laplace Theorem to higher order: n-th order difference quotients of the binomial probability distribution tend to the corresponding n-th order partial differential quotients of the Gaussian distribution. 2010 Mathematics Subject Classification 03H05, 39A10, 39A12, 60F05 (primary)