{
  "id": "2010.13526",
  "version": "v1",
  "published": "2020-10-20T07:09:06.000Z",
  "updated": "2020-10-20T07:09:06.000Z",
  "title": "$q$-Analogues of some supercongruences related to Euler numbers",
  "authors": [
    "Victor J. W. Guo"
  ],
  "comment": "12 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let $E_n$ be the $n$-th Euler number and $(a)_n=a(a+1)\\cdots (a+n-1)$ the rising factorial. Let $p>3$ be a prime. In 2012, Sun proved the that $$ \\sum^{(p-1)/2}_{k=0}(-1)^k(4k+1)\\frac{(\\frac{1}{2})_k^3}{k!^3} \\equiv p(-1)^{(p-1)/2}+p^3E_{p-3} \\pmod{p^4}, $$ which is a refinement of a famous supercongruence of Van Hamme. In 2016, Chen, Xie, and He established the following result: $$ \\sum_{k=0}^{p-1}(-1)^k (3k+1)\\frac{(\\frac{1}{2})_k^3}{k!^3} 2^{3k} \\equiv p(-1)^{(p-1)/2}+p^3E_{p-3} \\pmod{p^4}, $$ which was originally conjectured by Sun. In this paper we give $q$-analogues of the above two supercongruences by employing the $q$-WZ method. As a conclusion, we provide a $q$-analogue of the following supercongruence of Sun: $$ \\sum_{k=0}^{(p-1)/2}\\frac{(\\frac{1}{2})_k^2}{k!^2} \\equiv (-1)^{(p-1)/2}+p^2 E_{p-3} \\pmod{p^3}. $$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-20T07:09:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B65",
      "11A07",
      "33F10"
    ],
    "keywords": [
      "th euler number",
      "van hamme",
      "wz method",
      "refinement",
      "famous supercongruence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}