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.
\[
f(x+t,y)- f(x-t,y) = - i [f(x,y+t)- f(x,y-t)].    (E.1)
\]