{
  "id": "2001.08799",
  "version": "v1",
  "published": "2020-01-23T20:54:52.000Z",
  "updated": "2020-01-23T20:54:52.000Z",
  "title": "Characterizations of the Borel triangle and Borel polynomials",
  "authors": [
    "Paul Barry"
  ],
  "comment": "24 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We use Riordan array theory to give characterizations of the Borel triangle and its associated polynomial sequence. We show that the Borel polynomials are the moment sequence for a family of orthogonal polynomials whose coefficient array is a Riordan array. The role of the Catalan matrix in defining the Borel triangle is examined. We generalize the Borel triangle to a family of two parameter triangles. Generating functions are expressed as Jacobi continued fractions, as well as the zeros of appropriate quadratic expressions. The Borel triangle is exhibited as a Hadamard product of matrices. We investigate the reversions of the triangles studied. We introduce the notion of Fuss-Borel triangles and Fuss-Catalan triangles. We end with some remarks on the Catalan triangle.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-23T20:54:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B36",
      "11B83",
      "11C20",
      "33C45"
    ],
    "keywords": [
      "borel polynomials",
      "characterizations",
      "riordan array theory",
      "appropriate quadratic expressions",
      "fuss-catalan triangles"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}