A Pexiderized wavelike partial difference functional equation (Q2458382)

Let \((G,+)\) be a \(2\)-divisible abelian group in the sense that the equation \(2u=v\) is solvable in \(G\) and let \(f\) be a complex valued function on \(G \times G\). The authors show that the functional equation \[ f(x+t,y)+f(x-t,y)=f(x,y+t)+f(x,y-t) \tag{\(*\)} \] is equivalent to the functional equation \[ f(x+t,y+t)+f(x+t,y-t)+f(x-t,y+t)+f(x-t,y-t)=f(x,y+2t)+f(x,y-2t)+2f(x,y). \] They also find the general solution of the Pexider type generalization \[ f_1(x+t,y)+f_2(x-t,y)=f_3(x,y+t)+f_4(x,y-t) \] of the above equation. The equation \((*)\) was studied by \textit{J. A. Baker} [Can. Math. Bull. 12, 837--846 (1969; Zbl 0187.09103)] as well.