{
  "id": "1406.1589",
  "version": "v1",
  "published": "2014-06-06T05:58:04.000Z",
  "updated": "2014-06-06T05:58:04.000Z",
  "title": "On $t$-extensions of the Hankel determinants of certain automatic sequences",
  "authors": [
    "Hao Fu",
    "Guo-Niu Han"
  ],
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "In 1998, Allouche, Peyri\\`ere, Wen and Wen considered the Thue--Morse sequence, and proved that all the Hankel determinants of the period-doubling sequence are odd integral numbers. We speak of $t$-extension when the entries along the diagonal in the Hankel determinant are all multiplied by~$t$. Then we prove that the $t$-extension of each Hankel determinant of the period-doubling sequence is a polynomial in $t$, whose leading coefficient is the {\\it only one} to be an odd integral number. Our proof makes use of the combinatorial set-up developed by Bugeaud and Han, which appears to be very suitable for this study, as the parameter $t$ counts the number of fixed points of a permutation. Finally, we prove that all the $t$-extensions of the Hankel determinants of the regular paperfolding sequence are polynomials in $t$ of degree less than or equal to $3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-06T05:58:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A05",
      "05A10",
      "05A15",
      "05A19",
      "11B50",
      "11B65",
      "11B85",
      "15A15"
    ],
    "keywords": [
      "hankel determinant",
      "automatic sequences",
      "odd integral number",
      "period-doubling sequence",
      "polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.1589F"
    }
  }
}