{
  "id": "2003.10883",
  "version": "v1",
  "published": "2020-03-24T14:33:52.000Z",
  "updated": "2020-03-24T14:33:52.000Z",
  "title": "Some $q$-congruences arising from certain identities",
  "authors": [
    "Chen Wang",
    "He-Xia Ni"
  ],
  "comment": "7 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "In this paper, by constructing some identities, we prove some $q$-analogues of some congruences. For example, for any odd integer $n>1$, we show that \\begin{gather*} \\sum_{k=0}^{n-1} \\frac{(q^{-1};q^2)_k}{(q;q)_k} q^k \\equiv (-1)^{(n+1)/2} q^{(n^2-1)/4} - (1+q)[n] \\pmod{\\Phi_n(q)^2},\\\\ \\sum_{k=0}^{n-1}\\frac{(q^3;q^2)_k}{(q;q)_k} q^k \\equiv (-1)^{(n+1)/2} q^{(n^2-9)/4} + \\frac{1+q}{q^2}[n]\\pmod{\\Phi_n(q)^2}, \\end{gather*} where the $q$-Pochhanmmer symbol is defined by $(x;q)_0=1$ and $(x;q)_k = (1-x)(1-xq)\\cdots(1-xq^{k-1})$ for $k\\geq1$, the $q$-integer is defined by $[n]=1+q+\\cdots+q^{n-1}$ and $\\Phi_n(q)$ is the $n$-th cyclotomic polynomial. The $q$-congruences above confirm some recent conjectures of Gu and Guo.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-24T14:33:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "congruences arising",
      "identities",
      "th cyclotomic polynomial",
      "pochhanmmer symbol",
      "odd integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}