Grunwald-Letnikov differintegral

In mathematics, the combined differentiation/integration operator used in fractional calculus is called the "differintegral". It takes a few different forms, depending on context. The Grunwald-Letnikov differintegral has one of the simplest definitions, and is a commonly used form of the differintegral. It was introduced by Anton Karl Grünwald (1838-1920) from Prague, in 1867, and by Aleksey Vasilievich Letnikov (1837-1888) in Moscow in 1868.

It is a heuristic extension of the definition of the derivative:

:$f\text{'}(x)\; =\; lim\_\{h\; o\; 0\}\; frac\{f(x+h)-f(x)\}\{h\}$

Constructing the Grunwald-Letnikov differintegral

The formula for the derivative can be applied recursively to get higher-order derivatives.For example, the second-order derivative would be:

:$f"(x)\; =\; lim\_\{h\; o\; 0\}\; frac\{f\text{'}(x+h)-f\text{'}(x)\}\{h\}$

:$=\; lim\_\{h\_1\; o\; 0\}\; frac\{lim\_\{h\_2\; o\; 0\}\; frac\{f(x+h\_1+h\_2)-f(x+h\_1)\}\{h\_2\}-lim\_\{h\_2\; o\; 0\}\; frac\{f(x+h\_2)-f(x)\}\{h\_2\{h\_1\}$

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

:$=\; lim\_\{h\; o\; 0\}\; frac\{f(x+2h)-2f(x+h)+f(x)\}\{h^2\}$

In general, we have (see binomial coefficient):

:$d^n\; f(x)\; =\; lim\_\{h\; o\; 0\}\; frac\{sum\_\{0\; le\; m\; le\; n\}(-1)^m\; \{n\; choose\; m\}f(x+(n-m)h)\}\{h^n\}$

Formally, removing the restriction that "n" be a positive integer, we have:

:

This defines the Grunwald-Letnikov differintegral.

Another notation

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

:

This results in the expression:

:$mathbb\{D\}^q\; f(x)\; =\; lim\_\{h\; o\; 0\}frac\{Delta^q\_h\; f(x)\}\{h^q\}$