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  "body": "\n## Summary\n\nA sequence of orthogonal polynomials is a sequence $p_0, p_1, p_2, \\dots$ with $\\deg p_n = n$ that are pairwise orthogonal.\nOrthogonality refers to the inner product $\\int p q \\, d\\mu$ of a measure $\\mu$ on the real line.\nThey arise from the monomials $1, x, x^2, \\dots$ by Gram–Schmidt orthogonalization, and they are unique up to constant factors.\nThe classical families of Hermite, Laguerre, Jacobi, Legendre and Chebyshev polynomials are examples.\nThey satisfy a three-term recurrence, their zeros are real and interlace, and they are used in approximation, numerical integration, spectral theory and random matrices.\n\n## Explanations\n\n### L1 human\n\nTwo polynomials are orthogonal for a weight if the integral of their product against the weight is zero.\nStart with $1, x, x^2, \\dots$ and straighten them one at a time so that each is orthogonal to all earlier ones.\nThe result is a list of polynomials, one of each degree, that act like perpendicular axes in a space of functions.\nFor the plain weight on $[-1, 1]$ one gets the Legendre polynomials $1$, $x$, $\\frac12(3x^2 - 1)$, and so on.\n",
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