Why the following integral integrated by parts results in:
\begin{align} \sum_{\alpha_1+\dots+\alpha_d=m}\frac{m!}{\alpha_1!\dots\alpha_d!}\int_{-\infty}^{\infty}\dots\int_{-\infty}^{\infty}\left(\frac{\partial^mf}{\partial x_1^{\alpha_1}\dots\partial x_d^{\alpha_d}}\right)\left(\frac{\partial^mg}{\partial x_1^{\alpha_1}\dots\partial x_d^{\alpha_d}}\right)\prod_j dx_j\\ =(-1)^m\int_{-\infty}^{\infty}\dots\int_{-\infty}^{\infty}f\cdot \Delta^mg+c_{\infty} \end{align} where $c_{\infty}$ is due to the boundary values.