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  "body": "\n## Statement\n\nLet $\\mu$ be a positive [[Borel measure|DYR1QFNGNJ]] on $\\R$ with infinite support and with $\\int_\\R \\abs{x}^k\\, d\\mu(x) \u003c \\infty$ for every $k \\ge 0$.\nLet $\\ip{f}{g} = \\int_\\R f g\\, d\\mu$, and let $(p_n)_{n \\ge 0}$ be the monic orthogonal polynomials for $\\mu$, so that $\\deg p_n = n$, $p_n$ has leading coefficient $1$, and $\\ip{p_m}{p_n} = 0$ for $m \\ne n$.\nSet $p_{-1} = 0$.\nThen there are real numbers $a_n$ for $n \\ge 0$ and $b_n \u003e 0$ for $n \\ge 1$ such that, for every $n \\ge 0$,\n$$p_{n+1}(x) = (x - a_n)\\, p_n(x) - b_n\\, p_{n-1}(x),$$\nnamely $a_n = \\frac{\\ip{x p_n}{p_n}}{\\ip{p_n}{p_n}}$ and $b_n = \\frac{\\ip{p_n}{p_n}}{\\ip{p_{n-1}}{p_{n-1}}}$.\n\n## Remarks\n\nInfinite support makes $\\ip{\\cdot}{\\cdot}$ an [[inner product|0NB0AWGJBW]] on all polynomials, so the [[Gram–Schmidt process|0VT6P0GNQR]] applied to $1, x, x^2, \\dots$ gives the $p_n$.\nThe polynomial $x p_n - p_{n+1}$ has degree at most $n$, so it is a combination of $p_0, \\dots, p_n$.\nIts coefficient on $p_k$ is $\\frac{\\ip{x p_n}{p_k}}{\\ip{p_k}{p_k}} = \\frac{\\ip{p_n}{x p_k}}{\\ip{p_k}{p_k}}$, which is $0$ for $k \\le n - 2$ because $x p_k$ has degree less than $n$.\nThis key step uses that multiplication by $x$ is symmetric for $\\ip{\\cdot}{\\cdot}$.\nThe same symmetry gives $\\ip{x p_n}{p_{n-1}} = \\ip{p_n}{x p_{n-1}} = \\ip{p_n}{p_n}$, which yields the formula for $b_n$.\nExample: for the [[Hermite polynomials|MGACY98E9H]] made monic, $a_n = 0$ and $b_n = \\frac{n}{2}$, so $p_2 = x^2 - \\frac{1}{2}$.\n\n## Explanations\n\n### L1 human\n\nOrthogonal polynomials are not just any list of polynomials: each one is built from the previous two by a short rule.\nTo get the next one, multiply the current one by $x$, shift by a multiple of it, and subtract a multiple of the one before.\nThis makes them cheap to compute and gives them many special properties.\n\n### L2 human\n\nThe recurrence coefficients encode the measure: they form a symmetric tridiagonal Jacobi matrix whose spectral measure is $\\mu$, normalised to mass $1$, when $\\mu$ is determined by its moments.\nThe converse, that any such recurrence with $b_n \u003e 0$ comes from a measure, is Favard's theorem.\nThe zeros of $p_n$ are the eigenvalues of the $n \\times n$ truncated Jacobi matrix, which is the basis of the Golub–Welsch method for Gaussian quadrature.\nThe Christoffel–Darboux formula is a direct consequence of the recurrence.\n",
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