In mathematics, the combined differentiation/integration operator used in fractional calculus is called the differintegral, and it has a few different forms which are all equivalent, provided that they are initialized (used) properly.
It is noted:
The Grunwald-Letnikov differintegral is a commonly used form of the differintegral. It is defined via the fundamental theorem of calculus (see derivative):
Constructing the Grunwald-Letnikov differintegral
The formula for derivative can be applied recursively to get higher-order derivatives.
For example, the second-order derivative would be:
This is the Grunwald-Letnikov differintegral.