Grünwald–Letnikov derivative

Remarks

Example: for $\alpha = 1$, $\binom{1}{0} = \binom{1}{1} = 1$ and $\binom{1}{k} = 0$ for $k \ge 2$. The sum is $f(x) - f(x - h)$ once $h \le x - a$, so ${}^{GL}D_a^1 f(x)$ is the left derivative of $f$ at $x$, and it equals $f'(x)$ when $f$ is differentiable at $x$. More generally, for an integer $\alpha = n$ the sum is the $n$-th backward difference, and the limit is $f^{(n)}(x)$ when $f$ is $n$ times differentiable at $x$. Non-example: for $f(t) = t^{1/3}$ (the real cube root), $a = -1$, $x = 0$ and $\alpha = 1$, the quotient is $(0 - (-h)^{1/3})/h = h^{-2/3}$. It tends to $\infty$, so the limit does not exist in $\R$. The definition only sees values to the left of $x$: for $f(t) = \abs{t}$ at $x = 0$ it gives the one-sided value $-1$, although $f'(0)$ does not exist. For $n - 1 < \alpha < n$ and $f$ of class $C^n$ on $[a, x]$, the Grünwald–Letnikov derivative equals the Riemann–Liouville derivative of fractional calculus.