{
  "id": "2203.12901",
  "version": "v1",
  "published": "2022-03-24T07:38:22.000Z",
  "updated": "2022-03-24T07:38:22.000Z",
  "title": "Transcendence and continued fraction expansion of values of Hecke-Mahler series",
  "authors": [
    "Yann Bugeaud",
    "Michel Laurent"
  ],
  "comment": "29 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\theta$ and $\\rho$ be real numbers with $0 \\le \\theta, \\rho < 1$ and $\\theta$ irrational. We show that the Hecke-Mahler series $$ F_{\\theta, \\rho} (z_1, z_2) = \\sum_{k_1 \\ge 1} \\, \\sum_{k_2 = 1}^{\\lfloor k_1 \\theta + \\rho \\rfloor} \\, z_1^{k_1} z_2^{k_2}, $$ where $\\lfloor \\cdot \\rfloor$ denotes the integer part function, takes transcendental values at any algebraic point $(\\beta, \\alpha)$ with $0 < |\\beta|, |\\beta \\alpha^\\theta | < 1$. This extends earlier results of Mahler (1929) and Loxton and van der Poorten (1977), who settled the case $\\rho=0$. Furthermore, for positive integers $b$ and $a$, with $b \\ge 2$ and $a$ congruent to $1$ modulo $b-1$, we give the continued fraction expansion of the number $$ {(b-1)^2\\over b} F_{\\theta, \\rho} \\left({1\\over b}, {1\\over a}\\right)+{\\lfloor \\theta+\\rho\\rfloor(b-1)\\over b^2a}, $$ from which we derive a formula giving the irrationality exponent of $F_{\\theta, \\rho} (1/b, 1/a)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-24T07:38:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J04",
      "11J70",
      "11J81"
    ],
    "keywords": [
      "continued fraction expansion",
      "hecke-mahler series",
      "transcendence",
      "extends earlier results",
      "integer part function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}