Grünwald–Letnikov derivative

Definition

The Grünwald–Letnikov derivative is an attempt to define a generalization for the iterated derivative to a non-integer number of iterations. For a real number $q$, a function $f$, and a real number $h$, define:

$\underset{h}{\overset{q}{\mathrm{\Delta }}}f(x)={\sum }_{0\le m<\mathrm{\infty }}(-1{)}^m(\genfrac{}{}{0ex}{}{q}{m})f(x+(q-m)h)$

Then, we define:

${\mathbb{D}}^{q}f(x)=\underset{h\to 0}{lim}\frac{\underset{h}{\overset{q}{\mathrm{\Delta }}}f(x)}{h^q}$

See also

• Riemann-Louiville fractional integral (this also allows us to compute the Riemann-Liouville fractional derivative)