{
  "id": "2403.19503",
  "version": "v1",
  "published": "2024-03-28T15:34:53.000Z",
  "updated": "2024-03-28T15:34:53.000Z",
  "title": "Supercongruences involving Apéry-like numbers and Bernoulli numbers",
  "authors": [
    "Ji-Cai Liu"
  ],
  "comment": "18 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "For non-negative integers $n,r,s,t$, let $A_n^{(r,s,t)}=\\sum_{k=0}^n{n\\choose k}^r{n+k\\choose k}^s{2k\\choose n}^t$, which includes six Ap\\'ery-like numbers as special cases. We establish one step deep congruences of the Gauss congruences for $\\{A_n^{(r,s,t)}\\}_{n\\ge 0}$ in the form: \\begin{align*} A_{np}^{(r,s,t)}\\equiv A_n^{(r,s,t)}+p^3B_{p-3}\\mathcal{A}^{(r,s,t)}_n\\pmod{p^4}, \\end{align*} for all primes $p\\ge 5$ and all positive integers $n,r$ with $r\\ge 2$, where $\\mathcal{A}^{(r,s,t)}_n$ is independent of $p$ and $B_{p-3}$ is the $(p-3)$th Bernoulli numbers. This type of supercongruences for another two Ap\\'ery-like numbers are also established.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-28T15:34:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A07",
      "11B65",
      "11M41"
    ],
    "keywords": [
      "apéry-like numbers",
      "supercongruences",
      "step deep congruences",
      "th bernoulli numbers",
      "apery-like numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}