On a discrete version of the wave equation (Q2340856)

The author considers the wave equation \[ {\partial^2 u\over\partial t^2}= {\partial^2 u\over\partial x^2} \] and the discrete counterpart, that is, the partial difference equation \[ \underset{(x)}\Delta^2_1 u(x,y)= \underset{(y)}\Delta^2_1 u(x,y) \] and solves the equation \[ u(x+ 2,y)- 2u(x+1, y)= u(x,y+ 2)- 2u(x,y+ 1),\tag{1} \] at first on \(\mathbb Z^2\) and also on \(\mathbb R^2\). The author assumes that for the function \(u: \mathbb Z^2\to \mathbb C\), the functional equation (1) is fulfilled for any \((x,y)\in \mathbb Z^2\) and proves that, then and only then, there exist functions \(\alpha\) and \(\beta\) defined on \(\mathbb Z\) and with values in \(\mathbb C\) such that \(\beta\) is a generalized second-order polynomial and \[ (\forall(x, y)\in \mathbb Z^2)(u(x,y)= \alpha(x+ y)+\beta(x- y)). \] Moreover, the author supposes that the function \(u:\mathbb R^2\to \mathbb C\) satisfies the functional equation (1) for all \((x,y)\in \mathbb R^2\); then and only then, there exist functions \(\alpha\) and \(\beta\) defined on \(\mathbb R\) and with values in \(\mathbb C\) such that \(\beta\) is a generalized second-order polynomial, further, there is an antisymmetric and bi-additive function \(B:\mathbb R^2\to \mathbb C\) such that \[ (\forall(x,y)\in \mathbb R^2)(u(x,y)= \alpha(x+y)- \beta(x-y)+ B(x,y)) \] is fulfilled. In particular, if the continuous function \(u:\mathbb R^2\to \mathbb C\) satisfies the functional equation (1) for all \((x,y)\in \mathbb R^2\), then and only then, there exists a continuous function \(\alpha: \mathbb R\to \mathbb C\) and constants \(a\), \(b\), \(c\) such that \[ (\forall(x,y)\in \mathbb R^2)(u(x,y)= \alpha(x+ y)+ a(x-y)^2+ b(x-y)+ c) \] holds.