{
  "id": "2105.15049",
  "version": "v1",
  "published": "2021-05-31T15:32:58.000Z",
  "updated": "2021-05-31T15:32:58.000Z",
  "title": "Shifted sums of the Bernoulli numbers, reciprocity, and denominators",
  "authors": [
    "Bernd C. Kellner"
  ],
  "comment": "11 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We consider the numbers $\\mathcal{B}_{r,s} = (\\mathbf{B}+1)^r \\, \\mathbf{B}^s$ (in umbral notation $\\mathbf{B}^n = \\mathbf{B}_n$ with the Bernoulli numbers) that have a well-known reciprocity relation, which is frequently found in the literature and goes back to the 19th century. In a recent paper, self-reciprocal Bernoulli polynomials, whose coefficients are related to these numbers, appeared in the context of power sums and the so-called Faulhaber polynomials. The numbers $\\mathcal{B}_{r,s}$ can be recursively expressed by iterated sums and differences, so it is not obvious that these numbers do not vanish in general. As a main result among other properties, we show the non-vanishing of these numbers besides trivial cases. We further derive an explicit product formula for their denominators, which follows from a von Staudt--Clausen type relation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-31T15:32:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B68",
      "11B65"
    ],
    "keywords": [
      "bernoulli numbers",
      "shifted sums",
      "denominators",
      "von staudt-clausen type relation",
      "well-known reciprocity relation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}