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  "body": "\n## Definition\n\nLet $\\psi$ be the [[digamma function|HWRRXB0YXS]] and let $m \\ge 0$ be an integer.\nThe *polygamma function of order $m$* is the meromorphic function\n$$\\psi^{(m)}(z) = \\frac{d^m}{dz^m} \\psi(z) = \\frac{d^{m+1}}{dz^{m+1}} \\log \\Gamma(z), \\qquad z \\in \\C \\setminus \\set{0, -1, -2, \\ldots}.$$\n\n## Remarks\n\nThe series $\\psi(z) = -\\gamma + \\sum_{k=0}^\\infty \\left(\\frac{1}{k+1} - \\frac{1}{z+k}\\right)$, with $\\gamma$ the Euler–Mascheroni constant, can be differentiated term by term.\nFor $m \\ge 1$ this gives\n$$\\psi^{(m)}(z) = (-1)^{m+1}\\, m! \\sum_{k=0}^\\infty \\frac{1}{(z+k)^{m+1}} = (-1)^{m+1}\\, m!\\, \\zeta(m+1, z),$$\nwhere $\\zeta(s, z)$ is the [[Hurwitz zeta function|G4SW7HJG88]].\n\nExample: for $m = 2$ and $z = 1$ the series gives $\\psi^{(2)}(1) = -2 \\sum_{k \\ge 1} k^{-3} = -2\\zeta(3)$.\n\nNon-example of a valid argument: at $z = -n$ with $n \\ge 0$ an integer the term $k = n$ of the series is infinite.\nThere $\\psi^{(m)}$ has a pole of order $m + 1$.\n\nThe function $\\psi^{(0)} = \\psi$ is the digamma function and $\\psi^{(1)}$ is the [[trigamma function|5VGVPENA0F]].\nThe recurrence $\\psi^{(m)}(z+1) = \\psi^{(m)}(z) + (-1)^m\\, m!\\, z^{-m-1}$ follows by differentiating $\\psi(z+1) = \\psi(z) + 1/z$ $m$ times.\n",
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