{
  "aliases": null,
  "backlinks": [
    {
      "field": "specializes",
      "target": "5VGVPENA0F"
    },
    {
      "field": "defines",
      "target": "GRZKGHMNTG"
    }
  ],
  "body": "\n## Summary\n\nThe polygamma function of order $m$ is the $m$-th derivative of the [[digamma function|HWRRXB0YXS]], so the $(m+1)$-th derivative of $\\log \\Gamma$.\nFor $m \\ge 1$ it is a series of inverse powers, closely tied to the [[Hurwitz zeta function|G4SW7HJG88]].\nIts values at $1$ are, up to sign and a factorial, values $\\zeta(m+1)$ of the [[Riemann zeta function|1XWBY4N0VT]].\nThe case $m = 1$ is the [[trigamma function|5VGVPENA0F]].\n\n## Explanations\n\n### L1 human\n\nStart from the logarithm of the gamma function and differentiate it again and again.\nThe first derivative is the digamma function, the second is the trigamma function, and the later ones are the polygamma functions.\nFrom order one on, each of them is a simple sum such as $\\sum_{k \\ge 0} \\frac{1}{(z+k)^2}$.\nFor example, the trigamma function at $1$ is $1 + \\frac14 + \\frac19 + \\cdots = \\frac{\\pi^2}{6}$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "33B15"
    ],
    "name": "Polygamma function",
    "same_as": {
      "wikidata": "Q857956",
      "wikipedia": "Polygamma function"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Polygamma_function"
      }
    ],
    "tag": "3J3QGKPMZQ"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/3J3QGKPMZQ/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/3J3QGKPMZQ.md",
  "msc": [
    "33B15"
  ],
  "name": "Polygamma function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/3J3/3J3QGKPMZQ.md",
  "status": null,
  "tag": "3J3QGKPMZQ",
  "uri": "m2://3J3QGKPMZQ",
  "url": "https://tamnd.github.io/math2/wiki/polygamma-function/",
  "writing": {
    "tag": "3J3QGKPMZQ",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}