Grunwald-Letnikov differintegral

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:

and is most generally defined as:

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:

Assuming that the h 's converge symmetricly, this simplifies to:

In general, we have (see binomial coefficient):

If we remove the restriction that n must be a positive integer, we have:

This is the Grunwald-Letnikov differintegral.

A Simplier Expression

We may also write the expression more simply if we make the substitution:

This results in the expression: