{
  "id": "2009.09201",
  "version": "v1",
  "published": "2020-09-19T09:54:36.000Z",
  "updated": "2020-09-19T09:54:36.000Z",
  "title": "Inverse relations and reciprocity laws involving partial Bell polynomials and related extensions",
  "authors": [
    "Alfred Schreiber"
  ],
  "comment": "The article continues the research reported by the author in his work \"Multivariate Stirling polynomials of the first and second kind\", Discrete Mathematics 338 (2015), 2462-2484. Preprint version: arXiv:1311.5067",
  "categories": [
    "math.CO",
    "math.AC"
  ],
  "abstract": "The object of this paper is, in the main, twofold: Firstly, to develop an algebraic setting for dealing with Bell polynomials and related extensions. Secondly, based on the author's previous work on multivariate Stirling polynomials (arXiv:1311.5067), to present a number of new results affecting different types of inverse relationships, among these (1) the use of multivariable Lah polynomials for characterizing self-orthogonal families of polynomials that can be represented by Bell polynomials, (2) the introduction of `generalized Lagrange inversion polynomials' that invert functions characterized in a specific way by sequences of constants, (3) a general reciprocity theorem according to which, in particular, the partial Bell polynomials $B_{n,k}$ and their orthogonal companions $A_{n,k}$ belong to one single class of Stirling polynomials: $A_{n,k}=(-1)^{n-k}B_{-k,-n}$. Moreover, of some numerical statements (such as Stirling inversion, Schl\\\"omilch-Schl\\\"afli formulas) generalized polynomial versions are established, and a number of well-known theorems (Jabotinsky, Mullin-Rota, Melzak, Comtet) are given new proofs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-19T09:54:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A19",
      "11B73",
      "11B83",
      "05A15",
      "05E99",
      "11C08",
      "13F25",
      "13N15",
      "40E99",
      "46E25"
    ],
    "keywords": [
      "partial bell polynomials",
      "related extensions",
      "reciprocity laws",
      "general reciprocity theorem",
      "generalized lagrange inversion polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}