{
  "id": "1802.03266",
  "version": "v1",
  "published": "2018-02-09T14:13:09.000Z",
  "updated": "2018-02-09T14:13:09.000Z",
  "title": "Analysis of Summatory Functions of Regular Sequences: Transducer and Pascal's Rhombus",
  "authors": [
    "Clemens Heuberger",
    "Daniel Krenn",
    "Helmut Prodinger"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The summatory function of a $q$-regular sequence in the sense of Allouche and Shallit is analysed asymptotically. The result is a sum of periodic fluctuations for eigenvalues of absolute value larger than the joint spectral radius of the matrices of a linear representation of the sequence. The Fourier coefficients of the fluctuations are expressed in terms of residues of the corresponding Dirichlet generating function. A known pseudo Tauberian argument is extended in order to overcome convergence problems in Mellin--Perron summation. Two examples are discussed in more detail: The case of sequences defined as the sum of outputs written by a transducer when reading a $q$ary expansion of the input and the number of odd entries in the rows of Pascal's rhombus.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-09T14:13:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A16",
      "11A63",
      "68Q45",
      "68R05"
    ],
    "keywords": [
      "summatory function",
      "pascals rhombus",
      "regular sequence",
      "transducer",
      "pseudo tauberian argument"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}