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  "body": "\n## Definition\n\nLet $K$ be a [[field|04C5ZBG1WT]] and let $f$ be a [[function|0EPAT51DXQ]] from a subset of $K$ to $K$.\nLet $x_0, \\dots, x_n$ be pairwise distinct points of its domain.\nThe *divided differences* of $f$ are defined by recursion on the number of points:\n\n1. $f[x_i] = f(x_i)$ for each $i$;\n2. for $k \\ge 1$ and $0 \\le i \\le n - k$,\n$$f[x_i, \\dots, x_{i+k}] = \\frac{f[x_{i+1}, \\dots, x_{i+k}] - f[x_i, \\dots, x_{i+k-1}]}{x_{i+k} - x_i}.$$\n\n## Remarks\n\nThe value $f[x_0, \\dots, x_n]$ is a symmetric function of the points: it does not depend on their order.\nIt has the closed form\n$$f[x_0, \\dots, x_n] = \\sum_{j=0}^{n} \\frac{f(x_j)}{\\prod_{k \\ne j} (x_j - x_k)}.$$\nIt is the coefficient of $x^n$ in the [[interpolation polynomial|SS1YMS2NPC]] of $f$ at $x_0, \\dots, x_n$.\nFor a real function $f$ with enough derivatives, the definition extends to repeated points by continuity; for example $f[x, x] = f'(x)$.\n\nExample: for $f(x) = x^2$ and the points $0, 1, 3$: $f[0, 1] = 1$, $f[1, 3] = \\frac{9 - 1}{2} = 4$, and $f[0, 1, 3] = \\frac{4 - 1}{3 - 0} = 1$.\nThis is the leading coefficient of $x^2$, as expected.\n\nNon-example: the recursion cannot be applied to the list $1, 1$ of repeated points, because the denominator $x_1 - x_0$ is $0$.\n",
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