{
  "id": "1912.00765",
  "version": "v1",
  "published": "2019-11-29T14:48:05.000Z",
  "updated": "2019-11-29T14:48:05.000Z",
  "title": "$q$-Supercongruences modulo the fourth power of a cyclotomic polynomial via creative microscoping",
  "authors": [
    "Victor J. W. Guo"
  ],
  "comment": "13 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "By applying Chinese remainder theorem for coprime polynomials and the \"creative microscoping\" method recently introduced by the author and Zudilin, we establish parametric generalizations of three $q$-supercongruences modulo the fourth power of a cyclotomic polynomial. The original $q$-supercongruences then follow from these parametric generalizations by taking the limits as the parameter tends to $1$ (l'H\\^opital's rule is utilized here). In particular, we prove a complete $q$-analogue of the (J.2) supercongruence of Van Hamme and a complete $q$-analogue of a \"divergent\" Ramanujan-type supercongruence, thus confirming two recent conjectures of the author. We also put forward some related conjectures, including a $q$-supercongruence modulo the fifth power of a cyclotomic polynomial.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-29T14:48:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33D15",
      "11A07",
      "11B65"
    ],
    "keywords": [
      "cyclotomic polynomial",
      "supercongruences modulo",
      "fourth power",
      "creative microscoping",
      "parametric generalizations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}