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  "body": "\n## Summary\n\nThe [[orthogonal polynomials|8TXASARQN1]] of a measure on the real line depend only on its [[moments|01XAHPADBA]].\nGiven a moment sequence $(m_n)$ whose [[Hankel matrices|PQ1D8FQB3E]] are positive definite, the monic orthogonal polynomials are explicit quotients of determinants built from the $m_n$.\nThis link lets one pass between a sequence of numbers, a family of polynomials, a three-term recurrence and a measure.\nIt is a main tool in the solution of the [[Hamburger moment problem|TBNCFXTMMW]] and in the theory of Gaussian quadrature.\n\n## Explanations\n\n### L1 human\n\nThe moments $m_n = \\int x^n \\, d\\mu$ are all one needs to compute integrals of polynomials against $\\mu$.\nOrthogonality of polynomials is defined by such integrals, so the moments alone determine the orthogonal polynomials.\nOne can write each orthogonal polynomial as a determinant whose entries are moments, with the powers $1, x, \\dots, x^n$ in the last row.\nFor the uniform weight on $[-1, 1]$ the moments are $2, 0, \\tfrac{2}{3}, 0, \\tfrac{2}{5}, \\dots$, and the determinant formula gives $1$, $x$, $x^2 - \\tfrac13$, the monic Legendre polynomials.\n",
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