Discretisations of higher order and the theorems of

Abstract

We study discrete functions on equidistant and nonequidistant infinitesimal grids. We consider their difference quotients of higher order and give conditions for their nearequality to the corresponding derivatives. Important tools are nonstandard notions of regularity of higher order, and the formula of Fa`a di Bruno for higher order derivatives and a iscrete version of it. As an application of such transitions from the discrete to the continuous we extend the DeMoivreLaplace Theorem to higher order: nth order difference quotients of the binomial probability distribution tend to the corresponding nth order partial differential quotients of the Gaussian distribution.