{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $a$ and $b$ be complex numbers with $b \\notin \\set{0, -1, -2, \\dots}$.\nWrite $(x)_n = x (x + 1) \\cdots (x + n - 1)$ for the [[Pochhammer symbol|YZHM3HSM0F]], with $(x)_0 = 1$.\n*Kummer's confluent hypergeometric function* is the [[power series|0MZB06ST2B]]\n$$M(a, b, z) = {}_1F_1(a; b; z) = \\sum_{n = 0}^{\\infty} \\frac{(a)_n}{(b)_n} \\frac{z^n}{n!}, \\qquad z \\in \\C.$$\nMore generally, a *confluent hypergeometric function* is any solution of [[Kummer's equation|NT917KFR57]] $z w'' + (b - z) w' - a w = 0$.\n\n## Remarks\n\nThe ratio of consecutive terms is $\\frac{(a + n) z}{(b + n)(n + 1)}$, which tends to $0$, so the series converges for every $z$ and $M(a, b, \\cdot)$ is entire.\nInserting the series into Kummer's equation shows that $M(a, b, z)$ is its solution that is analytic at $0$ with $M(a, b, 0) = 1$.\nExample: $M(a, a, z) = e^z$, since $(a)_n / (a)_n = 1$.\nExample: $M(1, 2, z) = \\frac{e^z - 1}{z}$, since $\\frac{(1)_n}{(2)_n\\, n!} = \\frac{1}{(n + 1)!}$.\nNon-example: for $b = -1$ the series is not defined, since $(b)_2 = 0$ appears in a denominator.\nIf $a = -m$ with $m$ a nonnegative integer, the series stops and $M(-m, b, z)$ is a polynomial of degree $m$; for $b = \\alpha + 1$ it is a multiple of the generalized Laguerre polynomial $L_m^{(\\alpha)}(z)$.\nAnother notation is $\\Phi(a; b; z)$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "6AZ4ZBN1M8",
    "kind": "definition",
    "msc": [
      "33C15"
    ],
    "name": "Confluent hypergeometric function",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Confluent_hypergeometric_function"
      }
    ],
    "tag": "Y0BA4WG2W3",
    "uses": [
      "YZHM3HSM0F",
      "NT917KFR57",
      "0MZB06ST2B"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/Y0BA4WG2W3/",
  "links": [
    {
      "field": "defines",
      "target": "6AZ4ZBN1M8"
    },
    {
      "field": "uses",
      "target": "YZHM3HSM0F"
    },
    {
      "field": "uses",
      "target": "NT917KFR57"
    },
    {
      "field": "uses",
      "target": "0MZB06ST2B"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/Y0BA4WG2W3.md",
  "msc": [
    "33C15"
  ],
  "name": "Confluent hypergeometric function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Y0B/Y0BA4WG2W3.md",
  "status": null,
  "tag": "Y0BA4WG2W3",
  "uri": "m2://Y0BA4WG2W3",
  "url": "https://tamnd.github.io/math2/wiki/confluent-hypergeometric-function-definition/",
  "writing": {
    "tag": "Y0BA4WG2W3",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}