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  "body": "\n## Statement\n\nLet $M$ be an additive commutative monoid, let $G$ be an additive abelian group, let $h \\in M$, and let $f \\colon M \\to G$ be a function.\nLet $\\Delta_h^k f$ be the [[$k$-th forward difference|E38801KVJ1]] of $f$ with step $h$.\nThen for every $n \\in \\N$ and every $x \\in M$,\n$$f(x + nh) = \\sum_{k=0}^{n} \\binom{n}{k} \\Delta_h^k f(x),$$\nwhere $\\binom{n}{k}$ is the [[binomial coefficient|092GEF38YD]].\n\n## Remarks\n\nThe [[shift operator|91QMF4R1CY]] satisfies $E_h = I + \\Delta_h$, and $I$ commutes with $\\Delta_h$.\nSo the [[binomial theorem|0EHWPWGHB2]] gives $E_h^n = \\sum_k \\binom{n}{k} \\Delta_h^k$, which is the formula applied at $x$.\nExample: for $f(x) = x^2$, $h = 1$ and $x = 0$, the differences at $0$ are $f(0) = 0$, $\\Delta f(0) = 1$, $\\Delta^2 f(0) = 2$, so $f(n) = n + 2\\binom{n}{2} = n + n(n-1) = n^2$.\nLet $f \\colon \\R \\to \\R$ and $h \\ne 0$, and replace the top entry $n$ of $\\binom{n}{k}$ by a real number $s$.\nThis gives the Newton forward interpolation polynomial\n$$p(x_0 + sh) = \\sum_{k=0}^{n} \\binom{s}{k} \\Delta_h^k f(x_0), \\qquad \\binom{s}{k} = \\frac{s(s-1)\\cdots(s-k+1)}{k!}.$$\nIt is the unique polynomial of degree at most $n$ that agrees with $f$ at $x_0, x_0 + h, \\dots, x_0 + nh$.\n\n## Explanations\n\n### L1 human\n\nIf you know the value of a function at one point and all its differences there, you can rebuild its values further along.\nThis is like Taylor's formula, with differences in place of derivatives and binomial coefficients in place of $s^k / k!$.\nFor a polynomial only finitely many differences are nonzero, so the formula recovers it from a finite table.\n\n### L2 human\n\nThe formula and the [[expansion of iterated differences in shifts|24D1EGHP4G]] are inverse to each other: they express the binomial transform and its inverse.\nIn numerical analysis it gives the interpolation polynomial at equally spaced nodes, a special case of [[polynomial interpolation|SS1YMS2NPC]].\nLetting $n \\to \\infty$ with real $s$ leads to the [[Newton series|Y2HXSEA2EE]] of $f$.\nOver the $p$-adic integers, the same expansion with infinitely many terms is [[Mahler's theorem|D115PG3T7K]].\n",
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