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  "body": "\n## Definition\n\nLet $\\alpha \u003e 0$ be real and let $k \\ge 0$ be an integer.\nThe generalised [[binomial coefficient|0GBZM0MT0F]] is $\\binom{\\alpha}{k} = \\frac{\\alpha (\\alpha - 1) \\cdots (\\alpha - k + 1)}{k!}$, with $\\binom{\\alpha}{0} = 1$.\nLet $a \u003c x$ be real numbers and let $f \\colon [a, x] \\to \\R$.\nThe *Grünwald–Letnikov derivative* of order $\\alpha$ of $f$ at $x$, with lower terminal $a$, is the [[limit|0PKRYB8ATM]]\n$${}^{GL}D_a^\\alpha f(x) = \\lim_{h \\to 0^+} \\frac{1}{h^\\alpha} \\sum_{k=0}^{\\floor{(x - a)/h}} (-1)^k \\binom{\\alpha}{k} f(x - kh),$$\nwhen this limit exists.\n\n## Remarks\n\nExample: for $\\alpha = 1$, $\\binom{1}{0} = \\binom{1}{1} = 1$ and $\\binom{1}{k} = 0$ for $k \\ge 2$.\nThe 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 $x$.\nMore 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 $x$.\nNon-example: for $f(t) = t^{1/3}$ (the real cube root), $a = -1$, $x = 0$ and $\\alpha = 1$, the quotient is $(0 - (-h)^{1/3})/h = h^{-2/3}$.\nIt tends to $\\infty$, so the limit does not exist in $\\R$.\nThe 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.\nFor $n - 1 \u003c \\alpha \u003c n$ and $f$ of class $C^n$ on $[a, x]$, the Grünwald–Letnikov derivative equals the Riemann–Liouville derivative of [[fractional calculus|T73XK15SA2]].\n\n## Explanations\n\n### L1 human\n\nThe definition takes the backward difference quotients that give whole-order derivatives and lets the order be any positive number.\nThe weights $(-1)^k \\binom{\\alpha}{k}$ come from expanding $(1 - z)^\\alpha$ as a power series.\nTruncating the limit at a small step $h$ gives a practical numerical approximation.\n",
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