{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\beta \u003e 0$ and $0 \u003c c \u003c 1$ be real numbers, and let $n \\ge 0$ be an integer.\nWrite $(a)_k$ for the [[Pochhammer symbol|YZHM3HSM0F]].\nThe *Meixner polynomial* of degree $n$ is\n$$M_n(x; \\beta, c) = \\sum_{k=0}^{n} \\frac{(-n)_k \\, (-x)_k}{(\\beta)_k \\, k!} \\left( 1 - \\frac{1}{c} \\right)^k,$$\nthat is, $M_n(x; \\beta, c) = {}_2F_1\\left( -n, -x; \\beta; 1 - \\frac{1}{c} \\right)$, a terminating [[hypergeometric function|463X5V31D9]].\n\n## Remarks\n\nThey are orthogonal on the nonnegative integers:\n$$\\sum_{x=0}^{\\infty} \\frac{(\\beta)_x}{x!} c^x M_m(x; \\beta, c) M_n(x; \\beta, c) = \\frac{c^{-n} \\, n!}{(\\beta)_n (1 - c)^\\beta} \\delta_{mn}.$$\nExample: $M_1(x; \\beta, c) = 1 - \\frac{(1 - c) x}{\\beta c}$.\nCheck: the weights sum to $(1 - c)^{-\\beta}$, and their mean value of $x$ is $\\frac{\\beta c}{1 - c}$, so the weighted sum of $M_1$ is $(1 - c)^{-\\beta} (1 - 1) = 0$.\nNon-example: for $c \\ge 1$ the clause $0 \u003c c \u003c 1$ fails, and the weights $\\frac{(\\beta)_x}{x!} c^x$ do not have a finite sum, so the orthogonality relation has no meaning.\nLimit to Laguerre: $\\lim_{c \\to 1} M_n\\left( \\frac{x}{1 - c}; \\alpha + 1, c \\right) = \\frac{L_n^{(\\alpha)}(x)}{L_n^{(\\alpha)}(0)}$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "CEC3DESCFT",
    "kind": "definition",
    "msc": [
      "33C45",
      "42C05"
    ],
    "name": "Meixner polynomials",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Meixner_polynomials"
      }
    ],
    "tag": "88GQ3KHQ3Q",
    "uses": [
      "YZHM3HSM0F",
      "463X5V31D9"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/88GQ3KHQ3Q/",
  "links": [
    {
      "field": "defines",
      "target": "CEC3DESCFT"
    },
    {
      "field": "uses",
      "target": "YZHM3HSM0F"
    },
    {
      "field": "uses",
      "target": "463X5V31D9"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/88GQ3KHQ3Q.md",
  "msc": [
    "33C45",
    "42C05"
  ],
  "name": "Meixner polynomials",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/88G/88GQ3KHQ3Q.md",
  "status": null,
  "tag": "88GQ3KHQ3Q",
  "uri": "m2://88GQ3KHQ3Q",
  "url": "https://tamnd.github.io/math2/wiki/meixner-polynomials-definition/",
  "writing": {
    "tag": "88GQ3KHQ3Q",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}