Partial difference equations arising from the Cauchy-Riemann equations

Abstract

We consider some functional equations arising from the Cauchy-Riemann equations, and certain related functional equations. First we propose a new functional equation (E.1) below, over a 2-divisible Abelian group, which is a discrete version of the Cauchy-Riemann equations, and give the general solutions of (E.1). Next we study a functional equation which is equivalent to (E.1). Further we propose and solve partial difference-differential functional equations and nonsymmetric partial difference equations which are also arising from the Cauchy-Riemann equations.