{
  "id": "2211.15874",
  "version": "v1",
  "published": "2022-11-29T02:18:05.000Z",
  "updated": "2022-11-29T02:18:05.000Z",
  "title": "Some congruences involving generalized Bernoulli numbers and Bernoulli polynomials",
  "authors": [
    "Ni Li",
    "Rong Ma"
  ],
  "comment": "21pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $[x]$ be the integral part of $x$, $n>1$ be a positive integer and $\\chi_n$ denote the trivial Dirichlet character modulo $n$. In this paper, we use an identity established by Z. H. Sun to get congruences of $T_{m,k}(n)=\\sum_{x=1}^{[n/m]}\\frac{\\chi_n(x)}{x^k}\\left(\\bmod n^{r+1}\\right)$ for $r\\in \\{1,2\\}$, any positive integer $m $ with $n \\equiv \\pm 1 \\left(\\bmod m \\right)$ in terms of Bernoulli polynomials. As its an application, we also obtain some new congruences involving binomial coefficients modulo $n^4$ in terms of generalized Bernoulli numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-29T02:18:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B68",
      "11A07",
      "B.2"
    ],
    "keywords": [
      "generalized bernoulli numbers",
      "bernoulli polynomials",
      "congruences",
      "trivial dirichlet character modulo",
      "positive integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}