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  "body": "\n## Definition\n\nLet $I = (a, b)$ be an open interval with $-\\infty \\le a \u003c b \\le \\infty$, and let $w \\colon I \\to (0, \\infty)$ be continuous with $\\int_I \\abs{x}^k w(x)\\, dx \u003c \\infty$ for every $k \\ge 0$.\nA sequence $(p_n)_{n \\ge 0}$ of real polynomials with [[degree|0P4JRMSWNH]] $\\deg p_n = n$ is *orthogonal* for $w$ if $\\int_I p_m(x)\\, p_n(x)\\, w(x)\\, dx = 0$ for all $m \\ne n$.\nSuch a sequence is a family of *classical orthogonal polynomials* if an affine change of variable $x \\mapsto c x + d$ with $c \\ne 0$ does the following.\nIt turns the pair $(I, w)$, up to a constant factor in $w$, into one of these:\n\n1. (Jacobi) $I = (-1, 1)$ and $w(x) = (1 - x)^\\alpha (1 + x)^\\beta$ with $\\alpha, \\beta \u003e -1$.\n2. (Laguerre) $I = (0, \\infty)$ and $w(x) = x^\\alpha e^{-x}$ with $\\alpha \u003e -1$.\n3. (Hermite) $I = \\R$ and $w(x) = e^{-x^2}$.\n\n## Remarks\n\nThe integral $\\ip{f}{g} = \\int_I f g\\, w\\, dx$ is an [[inner product|0NB0AWGJBW]] on real polynomials, and an orthogonal sequence for $w$ is determined by $w$ up to a nonzero constant factor in each $p_n$.\nExample: the [[Chebyshev polynomials|M6EAR1JHPP]] $T_n$ are Jacobi with $\\alpha = \\beta = -\\frac{1}{2}$; for instance $\\int_{-1}^1 T_0 T_1 (1 - x^2)^{-1/2}\\, dx = \\int_{-1}^1 \\frac{x}{\\sqrt{1 - x^2}}\\, dx = 0$, since the integrand is odd.\nExample: the [[Hermite polynomials|MGACY98E9H]] are the Hermite case, and the weight $e^{-x^2/2}$ of the probabilists' version is the Hermite weight after the change $x \\mapsto x / \\sqrt{2}$.\nA non-example: the orthogonal polynomials for $w(x) = e^{-x^4}$ on $\\R$ exist, but they are not classical, since no affine change of variable turns $e^{-x^4}$ into one of the three weights.\nThe classical families are exactly the orthogonal families that are eigenfunctions of an operator $\\sigma(x) y'' + \\tau(x) y'$ with $\\deg \\sigma \\le 2$ and $\\deg \\tau = 1$.\nThey are also exactly those whose derivatives $(p_{n+1}')_{n \\ge 0}$ are again orthogonal, and those given by a Rodrigues formula.\n",
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