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  "body": "\n## Statement\n\nLet $\\mu$ be a positive [[Borel measure|DYR1QFNGNJ]] on $\\R$ with infinite support and all moments finite.\nLet $(p_n)_{n \\ge 0}$ be the orthonormal polynomials for $\\mu$, with $\\deg p_n = n$ and leading coefficient $k_n \u003e 0$.\nSo $\\int_\\R p_m p_n\\, d\\mu$ is $1$ if $m = n$ and $0$ otherwise.\nThen for every $n \\ge 0$ and all real $x \\ne y$,\n$$\\sum_{j=0}^{n} p_j(x)\\, p_j(y) = \\frac{k_n}{k_{n+1}} \\cdot \\frac{p_{n+1}(x)\\, p_n(y) - p_n(x)\\, p_{n+1}(y)}{x - y},$$\nand for every real $x$,\n$$\\sum_{j=0}^{n} p_j(x)^2 = \\frac{k_n}{k_{n+1}} \\left( p_{n+1}'(x)\\, p_n(x) - p_n'(x)\\, p_{n+1}(x) \\right).$$\n\n## Remarks\n\nFor orthonormal polynomials the three-term recurrence reads $x p_j = \\alpha_{j+1} p_{j+1} + \\beta_j p_j + \\alpha_j p_{j-1}$ with $\\alpha_j = \\frac{k_{j-1}}{k_j}$.\nMultiplying it at $x$ by $p_j(y)$, subtracting the same with $x$ and $y$ exchanged, and summing over $j$ makes the sum telescope to the right-hand side.\nThe second formula is the limit $y \\to x$ of the first.\nExample: let $n = 0$ and let $\\mu$ be a probability measure with mean $m$ and variance $s^2$.\nThen $p_0 = 1$ and $p_1 = \\frac{x - m}{s}$, and the right-hand side is $\\frac{1}{1/s} \\cdot \\frac{(x - m)/s - (y - m)/s}{x - y} = 1$, which matches the left-hand side.\nThe left-hand side $K_n(x, y)$ is the reproducing kernel of the polynomials of degree at most $n$ for the [[inner product|0NB0AWGJBW]] $\\int f g\\, d\\mu$.\n\n## Explanations\n\n### L1 human\n\nAdding up the products $p_j(x) p_j(y)$ for all degrees up to $n$ looks like a long computation.\nThe formula says that the sum collapses to an expression in only the last two polynomials.\nThis is the polynomial analogue of the closed form for the Dirichlet kernel in Fourier series.\n\n### L2 human\n\nThe kernel $K_n(x, y)$ gives the orthogonal projection onto polynomials of degree at most $n$, so the formula controls partial sums of orthogonal expansions.\nThe diagonal form shows $p_{n+1}' p_n - p_n' p_{n+1} \u003e 0$, which implies that the zeros of $p_n$ and $p_{n+1}$ are real, simple and interlace.\nThe numbers $\\frac{1}{K_n(x, x)}$ are the Christoffel function, whose values at the zeros of $p_{n+1}$ are the Gaussian quadrature weights.\nIn random matrix theory the kernel $K_n$ describes the correlations of eigenvalues of unitary invariant ensembles.\n",
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  "name": "Christoffel–Darboux formula",
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