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  "body": "\n## Description\n\nThe (physicists') Hermite polynomials are the polynomials $H_n \\in \\Z[x]$, $n \\ge 0$, given by\n$$H_n(x) = (-1)^n e^{x^2} \\frac{d^n}{dx^n} e^{-x^2}.$$\nEquivalently, $H_0 = 1$, $H_1 = 2x$ and $H_{n+1} = 2x \\, H_n - 2n \\, H_{n-1}$ for $n \\ge 1$.\nThe probabilists' Hermite polynomials $\\operatorname{He}_n$ use the weight $e^{-x^2/2}$ instead: $\\operatorname{He}_n(x) = (-1)^n e^{x^2/2} \\frac{d^n}{dx^n} e^{-x^2/2}$.\n\n## Remarks\n\nThe first values are $H_2 = 4x^2 - 2$, $H_3 = 8x^3 - 12x$, and $\\operatorname{He}_2 = x^2 - 1$, $\\operatorname{He}_3 = x^3 - 3x$.\nCheck for $H_2$ with the recursion: $2x \\cdot 2x - 2 \\cdot 1 \\cdot 1 = 4x^2 - 2$.\nThe polynomial $H_n$ has degree $n$ and leading coefficient $2^n$, while $\\operatorname{He}_n$ is monic.\nThe two families are related by $H_n(x) = 2^{n/2} \\operatorname{He}_n(\\sqrt{2} \\, x)$.\nThey are orthogonal on $\\R$: $\\int_{-\\infty}^{\\infty} H_m(x) H_n(x) e^{-x^2} \\, dx = \\sqrt{\\pi} \\, 2^n n! \\, \\delta_{mn}$.\nThey satisfy $H_n' = 2n \\, H_{n-1}$ and $\\operatorname{He}_n' = n \\operatorname{He}_{n-1}$.\nIn quantum mechanics, the eigenfunctions of the harmonic oscillator are $H_n(x) e^{-x^2/2}$ up to scaling.\n\n## Explanations\n\n### L1 human\n\nDifferentiate the bell curve $e^{-x^2}$ again and again.\nEach derivative is a polynomial times $e^{-x^2}$, and up to sign these polynomials are the Hermite polynomials.\nThey are the natural polynomials to use when the weight is a Gaussian bell curve.\n",
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