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  "body": "\n## Statement\n\nLet $\\nu \\in \\C$ and let $C_\\nu$ be any one of $J_\\nu$, $Y_\\nu$, $H^{(1)}_\\nu$, $H^{(2)}_\\nu$.\nThese are a [[Bessel function of the first kind|CZCMRDWDYW]], of the [[second kind|GRSNDJG363]], or a [[Hankel function|N3CC5DRGT1]].\nThen for $z \\in \\C \\setminus (-\\infty, 0]$:\n\n1. $C_{\\nu - 1}(z) + C_{\\nu + 1}(z) = \\frac{2\\nu}{z}\\, C_\\nu(z)$.\n2. $C_{\\nu - 1}(z) - C_{\\nu + 1}(z) = 2\\, C_\\nu'(z)$.\n\n## Remarks\n\nFor $J_\\nu$ both relations follow by comparing coefficients in the power series $J_\\nu(z) = \\sum_{m \\ge 0} \\frac{(-1)^m}{m!\\, \\Gamma(m + \\nu + 1)} \\left( \\frac{z}{2} \\right)^{2m + \\nu}$.\nEquivalent forms are $\\left( z^\\nu C_\\nu \\right)' = z^\\nu C_{\\nu-1}$ and $\\left( z^{-\\nu} C_\\nu \\right)' = -z^{-\\nu} C_{\\nu+1}$.\nFor $Y_\\nu$ they follow from the definition through $J_{\\pm\\nu}$ and a limit, and for the Hankel functions by linearity.\nExample: with $\\nu = 0$ the second relation and $J_{-1} = -J_1$ give $J_0' = -J_1$.\nExample: from $J_{1/2}(x) = \\sqrt{\\frac{2}{\\pi x}} \\sin x$ and $J_{-1/2}(x) = \\sqrt{\\frac{2}{\\pi x}} \\cos x$, the first relation with $\\nu = \\frac{1}{2}$ gives $J_{3/2}(x) = \\sqrt{\\frac{2}{\\pi x}} \\left( \\frac{\\sin x}{x} - \\cos x \\right)$.\n\n## Explanations\n\n### L1 human\n\nBessel functions of neighbouring orders are tied together by simple rules.\nIf one knows two consecutive orders, one gets all the others by the first rule.\nThe second rule gives the derivative of a Bessel function without any calculus on the function itself.\n\n### L2 human\n\nThe first relation is a three-term recurrence in the order, so tables and software compute $J_n(x)$ for many $n$ from a few starting values.\nForward use of it is unstable for $J_n$ when $n \u003e x$, and one runs it backward instead (Miller's algorithm).\nThe relations also show that Bessel functions of half-integer order are elementary, which gives the spherical Bessel functions.\nThe modified Bessel functions satisfy the same relations with some signs changed.\n",
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