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'(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'(x+h)-f'(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:

:$$mathbb{D}^q f(x) = lim_{h o 0} frac{1}{h^q}sum_{0 le m < infty}(-1)^m {q choose m}f(x+(q-m)h)

This defines the Grunwald-Letnikov differintegral.

Another notation

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

:$$Delta^q_h f(x) = sum_{0 le m < infty}(-1)^m {q choose m}f(x+(q-m)h)

This results in the expression:

:$$mathbb{D}^q f(x) = lim_{h o 0}frac{Delta^q_h f(x)}{h^q}