Higher-order finite difference

Definition

A higher-order finite difference for a function $f$ might refer to a function obtained by iterating one of the three standard forms of the finite difference: the forward difference, backward difference, or central difference. For a function $f$ at a point $x$ in the domain, and a positive integer $n$, we have three cases:

Name	Symbol	Expression
$n^{th}$ order forward difference	$\Delta _{h}^{n}[f](x)$	$\sum _{i=0}^{n}(-1)^{i}{\binom {n}{i}}f(x+(n-i)h)$
$n^{th}$ order backward difference	$\nabla _{h}^{n}[f](x)$	$\sum _{i=0}^{n}(-1)^{i}{\binom {n}{i}}f(x-ih)$
$n^{th}$ order central difference	$\delta _{h}^{n}[f](x)$	$\sum _{i=0}^{n}(-1)^{i}{\binom {n}{i}}f\left(x+\left({\frac {n}{2}}-i\right)h\right)$