Higher-order finite difference

Definition

A higher-order finite difference for a function $f$ might refer to a function obtained by iterating 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	$\underset{h}{\overset{n}{\mathrm{\Delta }}}[f](x)$	$\sum _{i=0}^n(-1{)}^i(\genfrac{}{}{0ex}{}{n}{i})f(x+(n-i)h)$
$n^{th}$ order backward difference	$\underset{h}{\overset{n}{\mathrm{\nabla }}}[f](x)$	$\sum _{i=0}^n(-1{)}^i(\genfrac{}{}{0ex}{}{n}{i})f(x-ih)$
$n^{th}$ order central difference	$\underset{h}{\overset{n}{\delta }}[f](x)$	$\sum _{i=0}^n(-1{)}^i(\genfrac{}{}{0ex}{}{n}{i})f(x+(\frac{n}{2}-i)h)$