{
  "id": "2001.08079",
  "version": "v1",
  "published": "2020-01-19T15:10:17.000Z",
  "updated": "2020-01-19T15:10:17.000Z",
  "title": "A family of $q$-congruences modulo the square of a cyclotomic polynomial",
  "authors": [
    "Victor J. W. Guo"
  ],
  "comment": "6 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Using Watson's terminating $_8\\phi_7$ transformation formula, we prove a family of $q$-congruences modulo the square of a cyclotomic polynomial, which were originally conjectured by the author and Zudilin [J. Math. Anal. Appl. 475 (2019), 1636--646]. As an application, we deduce two supercongruences modulo $p^4$ ($p$ is an odd prime) and their $q$-analogues. This also partially confirms a special case of Swisher's (H.3) conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-19T15:10:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33D15",
      "11A07",
      "11B65"
    ],
    "keywords": [
      "cyclotomic polynomial",
      "transformation formula",
      "supercongruences modulo",
      "odd prime",
      "special case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}