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  "body": "\n## Definition\n\n*Quantum calculus* is the study of [[functions|0EPAT51DXQ]] through the following two difference operators in place of the derivative.\n\n1. For a fixed $q \\in \\C \\setminus \\set{0, 1}$, the *$q$-derivative* of $f$ is $D_q f(x) = \\dfrac{f(qx) - f(x)}{(q - 1)x}$ for $x \\ne 0$; this is the [[q-derivative|PMJT3FZE0Y]].\n2. For a fixed $h \\in \\C \\setminus \\set{0}$, the *$h$-derivative* of $f$ is $D_h f(x) = \\dfrac{f(x + h) - f(x)}{h}$.\n\nThe corresponding antiderivatives are the [[Jackson integral|7EMPA6M8S7]] $\\int_0^x f(t) \\, d_q t = (1 - q) x \\sum_{k \\ge 0} q^k f(q^k x)$, for $0 \u003c \\abs{q} \u003c 1$ when the series converges, and the sum $h \\sum_{k=0}^{n-1} f(a + kh)$ over a grid of step $h$.\nThe part built on $D_q$ is called *$q$-calculus*, and the part built on $D_h$ is called *$h$-calculus*.\n\n## Remarks\n\nExample: for $f(x) = x^n$ with $n \\ge 1$, $D_q f(x) = \\dfrac{q^n x^n - x^n}{(q - 1)x} = [n]_q x^{n-1}$, where $[n]_q = 1 + q + \\cdots + q^{n-1}$.\n\nNon-example: the ordinary derivative $f'(x) = \\lim_{h \\to 0} (f(x + h) - f(x))/h$ is not an operator of quantum calculus, because it involves a limit rather than a fixed step $q$ or $h$.\nIt is the common limit of $D_q$ as $q \\to 1$ and of $D_h$ as $h \\to 0$ for differentiable $f$.\n\nThe $q$-derivative satisfies the product rule $D_q(fg)(x) = f(qx) D_q g(x) + g(x) D_q f(x)$.\nThe Jackson integral inverts $D_q$: if $F(x) = \\int_0^x f(t) \\, d_q t$ converges at $x$, then $D_q F(x) = f(x)$.\n",
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