It was introduced by Anton Karl Grünwald (1838–1920) from Prague, in 1867, and by Aleksey Vasilievich Letnikov (1837–1888) in Moscow in 1868.
For example, the second-order derivative would be: Assuming that the h 's converge synchronously, this simplifies to: which can be justified rigorously by the mean value theorem.
In general, we have (see binomial coefficient): Removing the restriction that n be a positive integer, it is reasonable to define: This defines the Grünwald–Letnikov derivative.
It can be shown that the equation may also be written as or removing the restriction that n must be a positive integer: 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:[1]