{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0578PA53VA"
    },
    {
      "field": "uses",
      "target": "0Z8PSSAVBA"
    }
  ],
  "body": "\n## Definition\n\nLet $R$ be a commutative ring and let $(a_n)_{n \\ge 0}$ be a sequence in $R$.\nThe ordinary generating function of $(a_n)$ is the [[formal power series|072W67KFYK]]\n$$A(x) = \\sum_{n \\ge 0} a_n x^n \\in R[\\![x]\\!].$$\n\n## Remarks\n\nExample: the sequence $1, 1, 1, \\dots$ has the generating function $\\sum_n x^n = \\frac{1}{1 - x}$, the [[geometric series|0R8JR0ARJ4]].\nOperations on sequences become operations on series. The [[product of power series|0H6KWAYS87]] corresponds to the convolution $c_n = \\sum_{i+j=n} a_i b_j$.\nThe [[exponential generating function|0KE6NBTVY8]] is a variant with the weights $x^n / n!$.\n\n## Explanations\n\n### L1 human\n\nA generating function stores a whole sequence as the coefficients of one power series.\nAlgebra with the series then gives facts about the sequence, such as recurrences or closed formulas.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0PHAC833JF",
    "kind": "definition",
    "msc": [
      "05A15"
    ],
    "name": "Generating function",
    "sources": [
      {
        "locator": "Ordinary generating function (OGF)",
        "url": "https://en.wikipedia.org/wiki/Generating_function"
      }
    ],
    "tag": "0HH6JX5WZZ",
    "uses": [
      "072W67KFYK"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0HH6JX5WZZ/",
  "links": [
    {
      "field": "defines",
      "target": "0PHAC833JF"
    },
    {
      "field": "uses",
      "target": "072W67KFYK"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0HH6JX5WZZ.md",
  "msc": [
    "05A15"
  ],
  "name": "Generating function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0HH/0HH6JX5WZZ.md",
  "status": null,
  "tag": "0HH6JX5WZZ",
  "uri": "m2://0HH6JX5WZZ",
  "url": "https://math2.tamnd.com/wiki/generating-function-definition/",
  "writing": {
    "tag": "0HH6JX5WZZ",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}