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  "body": "\n## Statement\n\nLet $P_n$ be the [[Legendre polynomial|QFYGJVA996]] of degree $n$ and let $J_0$ be the [[Bessel function of the first kind|CZCMRDWDYW]] of order $0$.\nThen\n$$\\lim_{n \\to \\infty} P_n\\left( \\cos \\frac{z}{n} \\right) = J_0(z),$$\nuniformly for $z$ in each bounded subset of $\\C$.\n\n## Remarks\n\nProof idea: write $P_n(\\cos \\theta) = {}_2F_1\\left( -n, n + 1; 1; \\sin^2 \\frac{\\theta}{2} \\right) = \\sum_{k=0}^{n} \\frac{(-n)_k (n + 1)_k}{(k!)^2} \\sin^{2k} \\frac{\\theta}{2}$.\nWith $\\theta = z/n$ and $k$ fixed, $(-n)_k (n + 1)_k \\sim (-1)^k n^{2k}$ and $\\sin^{2k} \\frac{z}{2n} \\sim \\left( \\frac{z}{2n} \\right)^{2k}$.\nSo the $k$-th term tends to $\\frac{(-1)^k}{(k!)^2} \\left( \\frac{z}{2} \\right)^{2k}$, the $k$-th term of the series of $J_0(z)$, and a domination argument allows the termwise limit.\nThe same holds for [[Jacobi polynomials|TRD7DSF2KR]]: $\\lim_{n \\to \\infty} n^{-\\alpha} P_n^{(\\alpha, \\beta)}\\left( \\cos \\frac{z}{n} \\right) = \\left( \\frac{z}{2} \\right)^{-\\alpha} J_\\alpha(z)$.\nA consequence via Hurwitz's theorem: if $\\theta_{n,k}$ is the $k$-th smallest zero of $\\theta \\mapsto P_n(\\cos \\theta)$ in $(0, \\pi)$, then $n \\theta_{n,k} \\to j_{0,k}$, the $k$-th positive zero of $J_0$.\n\n## Explanations\n\n### L1 human\n\nNear the end point $x = 1$, a Legendre polynomial of high degree oscillates faster and faster.\nIf one zooms in at the right scale, with the angle shrinking like $1/n$, the picture settles down to a fixed wave.\nThis limiting wave is the Bessel function $J_0$.\n\n### L2 human\n\nThe formula describes the behaviour of orthogonal polynomials at the edge of their interval, where the oscillatory asymptotics in the interior break down.\nIn the interior, the classical asymptotic formulas give cosine waves with slowly varying amplitude; near $x = \\pm 1$ Bessel functions take their place.\nIt explains why the zeros of $P_n$ closest to $1$, measured in the angle $\\theta$, are about $j_{0,k}/n$.\nIts versions for Jacobi and Laguerre polynomials, and their analogues for random matrices, describe the \"hard edge\" behaviour of eigenvalues.\n",
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