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  "body": "\n## Definition\n\nLet $M$ be an additive commutative monoid, let $G$ be an additive abelian group, and let $h \\in M$.\nThe *forward difference operator* with step $h$ sends a function $f \\colon M \\to G$ to the function $\\Delta_h f \\colon M \\to G$ given by\n$$\\Delta_h f(x) = f(x + h) - f(x) \\qquad (x \\in M).$$\nWhen $h = 1$, one writes $\\Delta$ for $\\Delta_1$.\nFor a sequence $u \\colon \\N \\to G$, this gives $(\\Delta u)_n = u_{n+1} - u_n$.\n\n## Remarks\n\nIt is a [[finite difference|7JSQZ126BP]] in the forward direction.\nThe operator $\\Delta_h$ is additive: $\\Delta_h(f + g) = \\Delta_h f + \\Delta_h g$.\nExample: for $f(x) = 2^x$ on $\\Z$ with $h = 1$, $\\Delta f(x) = 2^{x+1} - 2^x = 2^x$, so $f$ is a fixed point of $\\Delta$, like $e^x$ for the derivative.\nNon-example: $g(x) = f(x+1) - f(x-1)$ is not $\\Delta_h f$ for any $h$; it is a central difference with step $2$.\nWith the [[shift operator|MBK6AE6734]] $E_h f(x) = f(x + h)$ and the identity $I$, one has $\\Delta_h = E_h - I$.\n",
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