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  "body": "\n## Definition\n\nLet $\\alpha \u003e -1$ be a real number and let $n \\ge 0$ be an integer.\nThe *generalized Laguerre polynomial* of degree $n$ and parameter $\\alpha$ is\n$$L_n^{(\\alpha)}(x) = \\sum_{k=0}^{n} (-1)^k \\frac{\\Gamma(n + \\alpha + 1)}{(n - k)! \\, \\Gamma(k + \\alpha + 1)} \\frac{x^k}{k!},$$\nwhere $\\Gamma$ is the [[gamma function|0H5SYBSED9]].\nThe *Laguerre polynomial* of degree $n$ is $L_n = L_n^{(0)}$.\n\n## Remarks\n\nFor $\\alpha = 0$ the coefficient is a binomial coefficient: $L_n(x) = \\sum_{k=0}^{n} \\binom{n}{k} \\frac{(-x)^k}{k!}$.\nExample: $L_1^{(\\alpha)}(x) = 1 + \\alpha - x$ and $L_2(x) = 1 - 2x + \\frac{x^2}{2}$.\nRodrigues' formula: $L_n^{(\\alpha)}(x) = \\frac{x^{-\\alpha} e^x}{n!} \\frac{d^n}{dx^n} \\left( e^{-x} x^{n + \\alpha} \\right)$.\nOrthogonality: $\\int_0^\\infty L_m^{(\\alpha)}(x) L_n^{(\\alpha)}(x) \\, x^\\alpha e^{-x} \\, dx = \\frac{\\Gamma(n + \\alpha + 1)}{n!} \\delta_{mn}$.\nThe polynomial $L_n^{(\\alpha)}$ solves the [[Laguerre equation|FTGTM3JB1F]] $x y'' + (\\alpha + 1 - x) y' + n y = 0$.\nCheck for $n = 1$: $y = 1 + \\alpha - x$ gives $0 - (\\alpha + 1 - x) + (1 + \\alpha - x) = 0$.\nNon-example: the clause $\\alpha \u003e -1$ is needed for orthogonality, because for $\\alpha \\le -1$ the weight $x^\\alpha e^{-x}$ is not integrable near $0$.\n",
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