{
  "id": "1407.3343",
  "version": "v7",
  "published": "2014-07-12T05:01:47.000Z",
  "updated": "2014-08-20T08:28:55.000Z",
  "title": "Generalized $q$-Stirling numbers and normal ordering",
  "authors": [
    "Roberto B. Corcino",
    "Ken Joffaniel M. Gonzales",
    "Richell O. Celeste"
  ],
  "comment": "New section on q-Bell numbers added, extended to case $s\\in\\mathbb R$",
  "categories": [
    "math.CO"
  ],
  "abstract": "The normal ordering coefficients of strings consisting of $V,U$ which satisfy $UV=qVU+hV^s$ ($s\\in\\mathbb N$) are considered. These coefficients are studied in two contexts: first, as a multiple of a sequence satisfying a generalized recurrence, and second, as $q$-analogues of rook numbers under the row creation rule introduced by Goldman and Haglund. A number of properties are derived, including recurrences, expressions involving other $q$-analogues and explicit formulas. We also give a Dobinsky-type formula for the associated Bell numbers and the corresponding extension of Spivey's Bell number formula. The coefficients, viewed as rook numbers, are extended to the case $s\\in\\mathbb R$ via a modified rook model.",
  "revisions": [
    {
      "version": "v6",
      "updated": "2014-08-09T10:02:05.000Z",
      "abstract": "The normal ordering coefficients of strings consisting of $V,U$ which satisfy $UV=qVU+hV^s$ are considered. These coefficients are studied in two contexts: first, as special cases of a sequence satisfying a generalized recurrence, and second, as $q$-analogues of rook numbers under the model introduced by Goldman and Haglund. A number of properties are derived, including recurrences, expressions involving other $q$-analogues and explicit formulas. We also obtain a Dobinsky-type formula for the associated Bell numbers and give a rook theoretic proof of the corresponding extension of Spivey's Bell number formula.",
      "comment": "New section on q-Bell numbers added",
      "journal": null,
      "doi": null
    },
    {
      "version": "v7",
      "updated": "2014-08-20T08:28:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "11B65",
      "11B73"
    ],
    "keywords": [
      "stirling numbers",
      "spiveys bell number formula",
      "rook theoretic proof",
      "special cases"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.3343C"
    }
  }
}