Grünwald–Letnikov derivative
# Grünwald–Letnikov derivative

In mathematics, the Grünwald–Letnikov derivative is a basic extension of the derivative in fractional calculus that allows one to take the derivative a non-integer number of times. It was introduced by Anton Karl Grünwald (1838–1920) from Prague, in 1867, and by Aleksey Vasilievich Letnikov (1837–1888) in Moscow in 1868.

## Constructing the Grünwald–Letnikov derivative

The formula

    $f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}$

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

    ${\begin{aligned}f''(x)&=\lim _{h\to 0}{\frac {f'(x+h)-f'(x)}{h}}\\&=\lim _{h_{1}\to 0}{\frac {\lim \limits _{h_{2}\to 0}{\dfrac {f(x+h_{1}+h_{2})-f(x+h_{1})}{h_{2}}}-\lim \limits _{h_{2}\to 0}{\dfrac {f(x+h_{2})-f(x)}{h_{2}}}}{h_{1}}}\end{aligned}}$

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

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

which can be justified rigorously by the mean value theorem. In general, we have (see binomial coefficient):

    $f^{(n)}(x)=\lim _{h\to 0}{\frac {\sum \limits _{0\leq m\leq n}(-1)^{m}{n \choose m}f(x+(n-m)h)}{h^{n}}}$

Removing the restriction that n be a positive integer, it is reasonable to define:

    $\mathbb {D} ^{q}f(x)=\lim _{h\to 0}{\frac {1}{h^{q}}}\sum _{0\leq m<\infty }(-1)^{m}{q \choose m}f(x+(q-m)h).$

This defines the Grünwald–Letnikov derivative.

To simplify notation, we set:

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

So the Grünwald–Letnikov derivative may be succinctly written as:

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

### An alternative definition

In the preceding section, the general first principles equation for integer order derivatives was derived. It can be shown that the equation may also be written as

    $f^{(n)}(x)=\lim _{h\to 0}{\frac {(-1)^{n}}{h^{n}}}\sum _{0\leq m\leq n}(-1)^{m}{n \choose m}f(x+mh).$

or removing the restriction that n must be a positive integer:

    $\mathbb {D} ^{q}f(x)=\lim _{h\to 0}{\frac {(-1)^{q}}{h^{q}}}\sum _{0\leq m<\infty }(-1)^{m}{q \choose m}f(x+mh).$

This equation is called the reverse Grünwald–Letnikov derivative. If the substitution h → −h is made, the resulting equation is called the direct Grünwald–Letnikov derivative:

    $\mathbb {D} ^{q}f(x)=\lim _{h\to 0}{\frac {1}{h^{q}}}\sum _{0\leq m<\infty }(-1)^{m}{q \choose m}f(x-mh).$
