A Discrete Bochner–Martinelli Formula

Abstract

In the even dimensional case the discrete Dirac equation may be reduced to the so-called discrete isotonic Dirac system in which suitable Dirac operators appear from both sides in half the dimension. This is an appropriated framework for the development of a discrete Martinelli–Bochner formula for discrete holomorphic functions on the simplest of all graphs, the rectangular \({\mathbb{Z}^m}\) one. Two lower-dimensional cases are considered explicitly to illustrate the closed analogy with the theory of continuous variables and the developed discrete scheme.