{
  "id": "1506.08382",
  "version": "v1",
  "published": "2015-06-28T11:10:49.000Z",
  "updated": "2015-06-28T11:10:49.000Z",
  "title": "A combinatorial identity on Galton-Watson process",
  "authors": [
    "Linyuan Lu",
    "Arthur L. B. Yang"
  ],
  "comment": "9 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $f(m,c)=\\sum_{k=0}^{\\infty} (km+1)^{k-1} c^k e^{-c(km+1)/m} / (m^kk!)$. For any positive integer $m$ and positive real $c$, the identity $f(m,c)=f(1,c)^{1/m}$ arises in the random graph theory. In this paper, we present two elementary proofs of this identity: a pure combinatorial proof and a power-serial proof. We also proved that this identity holds for any positive reals $m$ and $c$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-28T11:10:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A20",
      "05C30",
      "05C80"
    ],
    "keywords": [
      "combinatorial identity",
      "galton-watson process",
      "positive real",
      "random graph theory",
      "pure combinatorial proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}