title: Grünwald–Letnikov derivative
text: # 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:
  
      $\mathbb {D} ^{q}f(x)=\lim _{h\to 0}{\frac {1}{h^{q}}}\sum _{0\leq m<\infty }(-1)^{m}{q \choose m}f(x-mh).$
  