---
tag: E38801KVJ1
kind: definition
name: Higher-order differences
msc: [39A70, 39A12]
defines: 3039GWNHWS
uses: [TXTR68V4R1, 2MM7AHFR2S]
sources:
  - url: https://en.wikipedia.org/wiki/Finite_difference
---

## Definition

Let $G$ and $H$ be additive abelian groups, let $h \in G$, and let $f \colon G \to H$ be a function.
Let $\Delta_h$ be the [[forward difference operator|8E0EBA8X07]] with step $h$.
The *forward differences of order $n$* of $f$, for $n \in \N$, are defined by induction:
$$\Delta_h^0 f = f, \qquad \Delta_h^{n+1} f = \Delta_h\left(\Delta_h^n f\right).$$
The backward differences $\nabla_h^n f$ of order $n$ are defined in the same way from the [[backward difference operator|MK5JGVDPS9]] $\nabla_h$.

## Remarks

Example: for $f(x) = x^2$ on $\R$, $\Delta_h f(x) = 2hx + h^2$ and $\Delta_h^2 f(x) = 2h(x + h) + h^2 - 2hx - h^2 = 2h^2$.
Non-example: $\left(\Delta_h f\right)^2$, the square of the first difference, is not $\Delta_h^2 f$; for $f(x) = x$ and $h = 1$ it is $1$, while $\Delta^2 f = 0$.
In terms of values, $\Delta_h^2 f(x) = f(x + 2h) - 2f(x + h) + f(x)$.
The two kinds are related by $\nabla_h^n f(x) = \Delta_h^n f(x - nh)$.