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  "body": "\n## Statement\n\nLet $\\sigma$ and $\\tau$ be real polynomials, not both zero, with $\\deg \\sigma \\le 2$ and $\\deg \\tau \\le 1$, and let $\\mathcal{D} = \\sigma(x) \\frac{d^2}{dx^2} + \\tau(x) \\frac{d}{dx}$.\nLet $(p_n)_{n \\ge 0}$ be real polynomials with $\\deg p_n = n$, each an eigenfunction of $\\mathcal{D}$: $\\mathcal{D} p_n = \\lambda_n p_n$ for some real [[eigenvalue|0BQAA2BK3Q]] $\\lambda_n$.\nAssume that $(p_n)$ is orthogonal for a weight.\nThat is, there are an open interval $I$ and a continuous $w \\colon I \\to (0, \\infty)$ with all moments finite such that $\\int_I p_m p_n\\, w\\, dx = 0$ for $m \\ne n$.\nThen, up to an affine change of variable and nonzero constant factors, $(p_n)$ is one of the [[classical orthogonal polynomial|47WJ34NC9K]] families: Jacobi, Laguerre or Hermite.\n\n## Remarks\n\nEvery operator $\\mathcal{D}$ of this form maps polynomials of degree at most $n$ to themselves.\nIn fact $\\mathcal{D} x^n = \\left( n(n-1)\\sigma_2 + n \\tau_1 \\right) x^n + \\text{lower terms}$, where $\\sigma_2$ and $\\tau_1$ are the coefficients of $x^2$ in $\\sigma$ and of $x$ in $\\tau$.\nThe proof sorts $\\sigma$ by its degree and its roots, up to affine changes.\nOrthogonality for a positive weight forces $w$ to solve the Pearson equation $(\\sigma w)' = \\tau w$, and only the Jacobi, Laguerre and Hermite cases give a positive weight with finite moments.\nExample: $\\sigma = 1$ and $\\tau = -2x$ give $\\mathcal{D} = \\frac{d^2}{dx^2} - 2x \\frac{d}{dx}$, with eigenfunctions the [[Hermite polynomials|MGACY98E9H]] and $\\lambda_n = -2n$; the check for $H_2 = 4x^2 - 2$ is $8 - 2x \\cdot 8x = -16x^2 + 8 = -4 H_2$.\nThe orthogonality hypothesis is necessary: $\\sigma = x^2$ and $\\tau = 2x + 2$ have the Bessel polynomials as eigenfunctions, which are not orthogonal for any positive weight on a real interval.\nWithout the orthogonality hypothesis, Bochner's list also contains the Bessel polynomials and the powers $x^n$, which are eigenfunctions of $x^2 \\frac{d^2}{dx^2}$.\n\n## Explanations\n\n### L1 human\n\nMany famous polynomial families solve a second-order differential equation, one polynomial for each degree.\nBochner asked which families can do this, and the answer is a short list.\nIf one also asks for orthogonality with a positive weight, only the Jacobi, Laguerre and Hermite families remain.\n\n### L2 human\n\nThis is the reason the classical families are singled out among all orthogonal polynomials.\nThe operator $\\mathcal{D}$ becomes symmetric for the inner product $\\int f g\\, w\\, dx$ when $(\\sigma w)' = \\tau w$, the Pearson equation for the weight, and $\\sigma w$ vanishes fast enough at the ends of $I$.\nThen $\\mathcal{D}$ is a [[Sturm–Liouville operator|7WV6B2502K]], and the $p_n$ are its eigenfunctions, orthogonal because they belong to different eigenvalues.\nDiscrete analogues with difference operators lead to the Hahn, Meixner, Krawtchouk and Charlier polynomials, and $q$-analogues to the Askey scheme.\n",
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