Is there some other way to characterize the functions $f:\mathbb Z\times \mathbb Z\to \mathbb Z$ which are expressible as
$$f(x,y)=g(x)+g(y)-g(x+y)$$
for some $g:\mathbb Z\to\mathbb Z$?


**Easy facts:** (1) $f$ must satisfy $f(x,y)=f(y,x)$ and $f(x,0)=g(0)$ for all $x$. (2) Not all functions $f$ are expressible, since for $x,y\in\{1,\ldots,n\}$ the number of choices of $f(x,y)$ the left grows quadratically in $n$ and the number of possible values of $g$ on which they depend, grows just linearly.



solvability of an elementary functional equation
The necessary and sufficient condition is that 
$$f(x,y+z)+f(y,z)=f(y,x+z)+f(x,z),\qquad\forall x,y,z\in\mathbb Z.\qquad (1)$$
On the one hand, this is obviously necessary. On the other hand, the function $g$, if it exists, is given by
$$g(k)=kg(1)-\sum\_{j=1}^{k-1}f(j,1).$$
Then $g$ is suitable if and only if
$$f(k,\ell)=\sum\_{j=1}^{k-1}f(j,1)-\sum\_{j=\ell}^{k+\ell-1}f(j,1),\qquad k,\ell\in\mathbb Z.$$
There remains to check that these relations are consequences of (1).


Note that (1) contains the fact that $x\mapsto f(x,0)$ is constant.


