{
  "id": "1812.11221",
  "version": "v1",
  "published": "2018-12-28T21:25:14.000Z",
  "updated": "2018-12-28T21:25:14.000Z",
  "title": "The Convergence Behavior of $q$-Continued Fractions on the Unit Circle",
  "authors": [
    "Douglas Bowman",
    "James Mc Laughlin"
  ],
  "comment": "11 pages. arXiv admin note: text overlap with arXiv:1812.10873",
  "journal": "The Ramanujan Journal 12 (2006), no. 2, 185--195",
  "doi": "10.1007/s11139-006-0072-4",
  "categories": [
    "math.NT"
  ],
  "abstract": "In a previous paper, we showed the existence of an uncountable set of points on the unit circle at which the Rogers-Ramanujan continued fraction does not converge to a finite value. In this present paper, we generalise this result to a wider class of $q$-continued fractions, a class which includes the Rogers-Ramanujan continued fraction and the three Ramanujan-Selberg continued fractions. We show, for each $q$-continued fraction, $G(q)$, in this class, that there is an uncountable set of points, $Y_{G}$, on the unit circle such that if $y \\in Y_{G}$ then $G(y)$ does not converge to a finite value. We discuss the implications of our theorems for the convergence of other $q$-continued fractions, for example the G\\\"ollnitz-Gordon continued fraction, on the unit circle.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-28T21:25:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A55"
    ],
    "keywords": [
      "unit circle",
      "convergence behavior",
      "rogers-ramanujan continued fraction",
      "finite value",
      "uncountable set"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}