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  "body": "\n## Definition\n\nLet $G$ and $H$ be additive abelian groups, let $h \\in G$, and let $f \\colon G \\to H$ be a function.\nLet $\\Delta_h$ be the [[forward difference operator|8E0EBA8X07]] with step $h$.\nThe *forward differences of order $n$* of $f$, for $n \\in \\N$, are defined by induction:\n$$\\Delta_h^0 f = f, \\qquad \\Delta_h^{n+1} f = \\Delta_h\\left(\\Delta_h^n f\\right).$$\nThe backward differences $\\nabla_h^n f$ of order $n$ are defined in the same way from the [[backward difference operator|MK5JGVDPS9]] $\\nabla_h$.\n\n## Remarks\n\nExample: for $f(x) = x^2$ on $\\R$, $\\Delta_h f(x) = 2hx + h^2$ and $\\Delta_h^2 f(x) = 2h(x + h) + h^2 - 2hx - h^2 = 2h^2$.\nNon-example: $\\left(\\Delta_h f\\right)^2$, the square of the first difference, is not $\\Delta_h^2 f$; for $f(x) = x$ and $h = 1$ it is $1$, while $\\Delta^2 f = 0$.\nIn terms of values, $\\Delta_h^2 f(x) = f(x + 2h) - 2f(x + h) + f(x)$.\nThe two kinds are related by $\\nabla_h^n f(x) = \\Delta_h^n f(x - nh)$.\n",
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