{
  "id": "1806.06432",
  "version": "v1",
  "published": "2018-06-17T19:19:14.000Z",
  "updated": "2018-06-17T19:19:14.000Z",
  "title": "Elements of Finite Order in the Riordan Group",
  "authors": [
    "Marshall M. Cohen"
  ],
  "comment": "19 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We consider elements $\\big(g(x), F(x)\\big)$ of finite order in the Riordan group over a field $\\cal F$ of characteristic $0$. We solve for all integers $n\\geq 2$ the two fundamental questions posed by L. Shapiro (2001) for the case $n = 2$ (\"involutions\"). Given a formal power series $F(x) = \\omega x + F_2x^2 + \\cdots$, and an integer $n\\geq 2$, Theorem 1 states exactly which $g(x)$ make $\\big(g(x), F(x))$ a Riordan element of order $n$. Theorem 2 classifies finite-order Riordan group elements up to conjugation. We then relate our work to papers (2008, 2013) of Cheon and Kim which motivated this paper. We supply a missing proof in the first paper and we solve the Open question in Section 2 of the second paper. Finally we show how this circle of ideas gives a new proof of C. Marshall's theorem (2017), which finds the unique $F(x)$, given bi-invertible $g(x)$, such that $\\big(g(x), F(x))$ is an involution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-17T19:19:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "20Hxx"
    ],
    "keywords": [
      "finite order",
      "classifies finite-order riordan group elements",
      "formal power series",
      "fundamental questions",
      "riordan element"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}