{
  "id": "1702.06519",
  "version": "v1",
  "published": "2017-02-21T18:47:35.000Z",
  "updated": "2017-02-21T18:47:35.000Z",
  "title": "A New Approach to the $r$-Whitney Numbers by Using Combinatorial Differential Calculus",
  "authors": [
    "José L. Ramírez",
    "Miguel A. Méndez"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "In the present article we introduce two new combinatorial interpretations of the $r$-Whitney numbers of the second kind obtained from the combinatorics of the differential operators associated to the grammar $G:=\\{ y\\rightarrow yx^{m}, x\\rightarrow x\\}$. By specializing $m=1$ we obtain also a new combinatorial interpretation of the $r$-Stirling numbers of the second kind. Again, by specializing to the case $r=0$ we introduce a new generalization of the Stirling number of the second kind and through them a binomial type family of polynomials that generalizes Touchard's. Moreover, we show several well-known identities involving the $r$-Dowling polynomials and the $r$-Whitney numbers using the combinatorial differential calculus. Finally we prove that the $r$-Dowling polynomials are a Sheffer family relative to the generalized Touchard binomial family, study their umbral inverses, and introduce $[m]$-Stirling numbers of the first kind. From the relation between umbral calculus and the Riordan matrices we give several new combinatorial identities involving the $r$-Whitney number of both kinds, Bernoulli and Euler polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-21T18:47:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B83",
      "11B73",
      "05A15",
      "05A19"
    ],
    "keywords": [
      "combinatorial differential calculus",
      "whitney number",
      "second kind",
      "stirling number",
      "combinatorial interpretation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}