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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $L_n$ be the [[Laguerre polynomial|WAMRHWW82R]] of degree $n$, normalised by $L_n(0) = 1$.\nLet $x_1, \\ldots, x_n$ be the zeros of $L_n$; they are simple and lie in $(0, \\infty)$.\nSet\n$$w_i = \\frac{x_i}{(n+1)^2 \\, L_{n+1}(x_i)^2} = \\frac{1}{x_i \\, L_n'(x_i)^2}, \\qquad i = 1, \\ldots, n.$$\nThen $w_i \u003e 0$ for each $i$, and for every polynomial $p$ of degree at most $2n - 1$,\n$$\\int_{0}^{\\infty} e^{-x} p(x) \\, dx = \\sum_{i=1}^{n} w_i \\, p(x_i).$$\n\n## Remarks\n\nThis is the [[Gaussian quadrature|FP37BTP93K]] rule for the weight $e^{-x}$ on $[0, \\infty)$, whose orthogonal polynomials are the $L_n$; the proof is the general one, by division by $L_n$ and orthogonality.\nExample: for $n = 2$, $L_2(x) = \\frac12 (x^2 - 4x + 2)$ has zeros $2 \\pm \\sqrt{2}$ and $L_2'(x) = x - 2 = \\pm\\sqrt{2}$ there, so $w = \\frac{1}{2(2 \\pm \\sqrt{2})} = \\frac{2 \\mp \\sqrt{2}}{4}$.\nCheck: the weights add up to $1 = \\int_0^\\infty e^{-x} dx$, and $\\sum_i w_i x_i^2 = \\frac{2 + \\sqrt{2}}{2} + \\frac{2 - \\sqrt{2}}{2} = 2 = \\int_0^\\infty x^2 e^{-x} dx$.\nThe degree $2n - 1$ cannot be raised: for $p = L_n^2$ the rule gives $0$ and the integral is positive.\nWith the generalized Laguerre polynomials $L_n^{(\\alpha)}$, $\\alpha \u003e -1$, the same construction gives rules for the weight $x^{\\alpha} e^{-x}$.\n\n## Explanations\n\n### L1 human\n\nIntegrals from $0$ to infinity with a decaying factor $e^{-x}$ appear in physics, probability and the Laplace transform.\nGauss–Laguerre quadrature replaces such an integral by a weighted sum of a few values of the remaining factor at special points.\nWith $n$ points it integrates every polynomial of degree up to $2n - 1$ exactly.\n\n### L2 human\n\nTo integrate $\\int_0^\\infty g(x)\\,dx$, one writes $g(x) = e^{-x} \\big(e^x g(x)\\big)$ and applies the rule to $f(x) = e^x g(x)$.\nThis works well when $f$ is smooth and grows at most like a polynomial.\nThe rule is the half-line companion of [[Gauss–Hermite quadrature|89S5KGQEMJ]] on the whole line and of Gauss–Legendre quadrature on a bounded interval.\n",
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