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  "body": "\n## Statement\n\nLet $(I, w, \\sigma)$ be one of the following triples:\n\n1. $I = (-1, 1)$, $w(x) = (1 - x)^\\alpha (1 + x)^\\beta$ with $\\alpha, \\beta \u003e -1$, and $\\sigma(x) = 1 - x^2$.\n2. $I = (0, \\infty)$, $w(x) = x^\\alpha e^{-x}$ with $\\alpha \u003e -1$, and $\\sigma(x) = x$.\n3. $I = \\R$, $w(x) = e^{-x^2}$, and $\\sigma(x) = 1$.\n\nFor $n \\ge 0$ let\n$$q_n(x) = \\frac{1}{w(x)}\\, \\frac{d^n}{dx^n} \\left( w(x)\\, \\sigma(x)^n \\right), \\qquad x \\in I.$$\nThen $q_n$ is a polynomial of [[degree|0P4JRMSWNH]] exactly $n$, and $\\int_I q_m(x)\\, q_n(x)\\, w(x)\\, dx = 0$ for $m \\ne n$.\nSo $(q_n)$ is, up to constant factors, the family of [[classical orthogonal polynomials|47WJ34NC9K]] for $w$.\n\n## Remarks\n\nThat $q_n$ is a polynomial follows by induction: each derivative of $w \\sigma^n$ is $w \\sigma^{n-k}$ times a polynomial of degree at most $k$.\nThis holds because $\\sigma w' / w$ is a polynomial of degree at most $1$.\nFor orthogonality, take $m \u003c n$ and integrate $\\int_I x^m\\, \\frac{d^n}{dx^n}(w \\sigma^n)\\, dx$ by parts $n$ times.\nThe boundary terms vanish because $w \\sigma^n$ and its first $n - 1$ derivatives vanish at the ends of $I$, and the final integrand contains $\\frac{d^n}{dx^n} x^m = 0$.\nThe usual normalisations are $P_n^{(\\alpha,\\beta)} = \\frac{(-1)^n}{2^n n!}\\, q_n$ (Jacobi), $L_n^{(\\alpha)} = \\frac{1}{n!}\\, q_n$ (Laguerre), and $H_n = (-1)^n q_n$ (Hermite).\nExample: in the Hermite case with $n = 2$, $\\frac{d^2}{dx^2} e^{-x^2} = (4x^2 - 2) e^{-x^2}$, so $q_2 = 4x^2 - 2 = H_2$.\nExample: in the Jacobi case with $\\alpha = \\beta = 0$, the formula gives the Legendre polynomials $P_n(x) = \\frac{1}{2^n n!} \\frac{d^n}{dx^n} (x^2 - 1)^n$.\n\n## Explanations\n\n### L1 human\n\nEach classical family of orthogonal polynomials comes from one recipe.\nMultiply the weight by a power of a simple polynomial, differentiate $n$ times, and divide by the weight again.\nFor the Legendre polynomials this means differentiating $(x^2 - 1)^n$ exactly $n$ times.\nThe formula makes orthogonality almost automatic, through integration by parts.\n\n### L2 human\n\nRodrigues' formula is the quickest way to compute norms, leading coefficients and generating functions of the classical families.\nWith Cauchy's integral formula it gives contour integral representations, from which generating functions such as $\\sum_n P_n(x)\\, t^n = (1 - 2xt + t^2)^{-1/2}$ follow.\nThe existence of such a formula characterises the classical families among orthogonal polynomials, together with Bochner's characterisation by differential equations.\nThe polynomial $\\sigma$ is the same one that appears in the second-order differential equation $\\sigma y'' + \\tau y' + \\lambda_n y = 0$ satisfied by $q_n$.\n",
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