{
  "id": "1812.06324",
  "version": "v1",
  "published": "2018-12-15T17:13:56.000Z",
  "updated": "2018-12-15T17:13:56.000Z",
  "title": "Some $q$-supercongruences from transformation formulas for basic hypergeometric series",
  "authors": [
    "Victor J. W. Guo",
    "Michael J. Schlosser"
  ],
  "comment": "39 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Several new $q$-supercongruences are obtained using transformation formulas for basic hypergeometric series, together with various techniques such as suitably combining terms, and creative microscoping, a method recently developed by the first author in collaboration with Wadim Zudilin. More concretely, the results in this paper include $q$-analogues of supercongruences (referring to $p$-adic identities remaining valid for some higher power of $p$) established by Long, by Long and Ramakrishna, and several other $q$-supercongruences. The six basic hypergeometric transformation formulas which are made use of are Watson's transformation, a quadratic transformation of Rahman, a cubic transformation of Gasper and Rahman, a quartic transformation of Gasper and Rahman, a double series transformation of Ismail, Rahman and Suslov, and a new transformation formula for a nonterminating very-well-poised ${}_{12}\\phi_{11}$ series. Also, the nonterminating $q$-Dixon summation formula is used. A special case of the new ${}_{12}\\phi_{11}$ transformation formula is further utilized to obtain a generalization of Rogers' linearization formula for the continuous $q$-ultraspherical polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-15T17:13:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33D15",
      "11A07",
      "11F33",
      "33D45"
    ],
    "keywords": [
      "basic hypergeometric series",
      "supercongruences",
      "basic hypergeometric transformation formulas",
      "dixon summation formula",
      "adic identities remaining valid"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}