---
tag: 4VE006XQ33
kind: definition
name: Grünwald–Letnikov derivative
msc: [26A33]
defines: M9Z8VM4K1Z
uses: [0GBZM0MT0F, 0PKRYB8ATM]
sources:
  - url: https://en.wikipedia.org/wiki/Gr%C3%BCnwald%E2%80%93Letnikov_derivative
---

## Definition

Let $\alpha > 0$ be real and let $k \ge 0$ be an integer.
The generalised [[binomial coefficient|0GBZM0MT0F]] is $\binom{\alpha}{k} = \frac{\alpha (\alpha - 1) \cdots (\alpha - k + 1)}{k!}$, with $\binom{\alpha}{0} = 1$.
Let $a < x$ be real numbers and let $f \colon [a, x] \to \R$.
The *Grünwald–Letnikov derivative* of order $\alpha$ of $f$ at $x$, with lower terminal $a$, is the [[limit|0PKRYB8ATM]]
$${}^{GL}D_a^\alpha f(x) = \lim_{h \to 0^+} \frac{1}{h^\alpha} \sum_{k=0}^{\floor{(x - a)/h}} (-1)^k \binom{\alpha}{k} f(x - kh),$$
when this limit exists.

## Remarks

Example: for $\alpha = 1$, $\binom{1}{0} = \binom{1}{1} = 1$ and $\binom{1}{k} = 0$ for $k \ge 2$.
The sum is $f(x) - f(x - h)$ once $h \le x - a$, so ${}^{GL}D_a^1 f(x)$ is the left derivative of $f$ at $x$, and it equals $f'(x)$ when $f$ is differentiable at $x$.
More generally, for an integer $\alpha = n$ the sum is the $n$-th backward difference, and the limit is $f^{(n)}(x)$ when $f$ is $n$ times differentiable at $x$.
Non-example: for $f(t) = t^{1/3}$ (the real cube root), $a = -1$, $x = 0$ and $\alpha = 1$, the quotient is $(0 - (-h)^{1/3})/h = h^{-2/3}$.
It tends to $\infty$, so the limit does not exist in $\R$.
The definition only sees values to the left of $x$: for $f(t) = \abs{t}$ at $x = 0$ it gives the one-sided value $-1$, although $f'(0)$ does not exist.
For $n - 1 < \alpha < n$ and $f$ of class $C^n$ on $[a, x]$, the Grünwald–Letnikov derivative equals the Riemann–Liouville derivative of [[fractional calculus|T73XK15SA2]].

## Explanations

### L1 human

The definition takes the backward difference quotients that give whole-order derivatives and lets the order be any positive number.
The weights $(-1)^k \binom{\alpha}{k}$ come from expanding $(1 - z)^\alpha$ as a power series.
Truncating the limit at a small step $h$ gives a practical numerical approximation.