Forward difference operator ($\Delta$)

The forward difference of a sequence or function on the integers is

On an $N$-pixel line with last forward difference zero, $D^{\ast }p$ has first entry $-p_1$, interior entries $p_{i-1}-p_i$, and last entry $p_{N-1}$, with $D^{\ast }=0$ for $N=1$. On a square grid add these expressions along rows and columns. Unused last-edge components contribute nothing. The identity $⟨Du,p⟩=⟨u,D^{\ast }p⟩$ fixes every boundary sign. On the unscaled square grid, $\Vert D{\Vert }^2=8{\cos }^2(\pi /(2N))\le 8$ for $N>1$.

A discrete antiderivative of $f$ is a function $F$ satisfying $\Delta F=f$. Pascal's identity gives

If ${\Delta }^{k+1}f=0$, then $f$ is an integer linear combination of $1,\left(\genfrac{}{}{0ex}{}{n}{1}\right),\dots ,\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ whenever $f$ is integer-valued. Successive differences recover the coefficients.