I derived this by repeatedly using the second-order finite difference formula for the first derivative.
Two questions:
1- ) Is the following finite difference formula correct?
2- ) If so, is the order of the error correct? The way I see it, we are applying the second-order formula for the first derivative again and again, so the order shouldn't change. Basically errors of order $O(\Delta^2 y)$ and $O(\Delta^2 x)$ keep adding up.
$\frac{\partial^4 f}{\partial x^2 \partial y^2} = \frac{f_{i+2,j+2} - 2f_{i,j+2} + f_{i-2,j+2} - 2f_{i+2,j} + 4f_{i,j} - 2f_{i-2,j} + f_{i+2,j-2} - 2f_{i,j-2} + f_{i-2,j-2}}{16 \Delta^2 y \Delta^2 x} + O(\Delta^2 y) + O(\Delta^2 x)$