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:

$\Delta _{h}^{q}f(x)=\sum _{0\leq m<\infty }(-1)^{m}{q \choose m}f(x+(q-m)h)$

Then, we define:

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

See also

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