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  "body": "\n## Statement\n\nLet $n$ be an integer and let $J_k$ denote the [[Bessel function of the first kind|CZCMRDWDYW]] of integer order $k$.\nFor all $x, y \\in \\C$,\n$$J_n(x + y) = \\sum_{k=-\\infty}^{\\infty} J_k(x)\\, J_{n-k}(y),$$\nwhere the series converges absolutely.\n\n## Remarks\n\nThe proof uses the [[generating function|0PHAC833JF]] $G(z, t) = \\exp\\left( \\frac{z}{2}\\left( t - \\frac{1}{t} \\right) \\right) = \\sum_{k \\in \\Z} J_k(z)\\, t^k$.\nSince $G(x + y, t) = G(x, t)\\, G(y, t)$, multiplying the two Laurent series in $t$ and comparing the coefficients of $t^n$ gives the formula.\nExample: with $n = 0$ and $y = -x$, one gets $1 = J_0(0) = \\sum_{k} J_k(x) J_{-k}(-x) = \\sum_k J_k(x)^2$, using $J_{-k}(-x) = J_k(x)$; so $J_0(x)^2 + 2\\sum_{k \\ge 1} J_k(x)^2 = 1$.\nThis identity shows $\\abs{J_0(x)} \\le 1$ and $\\abs{J_k(x)} \\le \\frac{1}{\\sqrt{2}}$ for $k \\ge 1$ and real $x$.\nThe formula as written is for integer orders; for non-integer orders there are related addition theorems of Graf and Gegenbauer.\n\n## Explanations\n\n### L1 human\n\nThe value of a Bessel function at a sum $x + y$ is built from its values at $x$ and at $y$.\nThis is the Bessel analogue of the rule $e^{x+y} = e^x e^y$, and indeed it comes from that rule applied to the generating function.\n\n### L2 human\n\nThe theorem is a convolution identity: the sequence $(J_k(x + y))_k$ is the convolution of $(J_k(x))_k$ and $(J_k(y))_k$.\nIn terms of Fourier series, it says that $e^{i(x+y)\\sin\\theta} = e^{ix\\sin\\theta} e^{iy\\sin\\theta}$, read coefficient by coefficient.\nIts consequence $\\sum_k J_k(x)^2 = 1$ gives uniform bounds on Bessel functions of integer order on the real line.\n",
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