{
  "id": "1601.02192",
  "version": "v1",
  "published": "2016-01-10T09:37:28.000Z",
  "updated": "2016-01-10T09:37:28.000Z",
  "title": "Some results associated with Bernoulli and Euler numbers with applications",
  "authors": [
    "C. -P. Chen",
    "R. B. Paris"
  ],
  "comment": "15 pages, 0 figures",
  "categories": [
    "math.CA"
  ],
  "abstract": "In this paper, we present series representations of the remainders in the expansions for $2/(e^t+1)$, $\\mbox{sech} t$ and $\\coth t$. For example, we prove that for $t > 0$ and $N\\in\\mathbb{N}:=\\{1, 2, \\ldots\\}$, \\[\\mbox{sech}\\, t=\\sum_{j=0}^{N-1}\\frac{E_{2j}}{(2j)!}t^{2j}+R_N(t) \\] with \\[ R_N(t)=\\frac{(-1)^{N}2t^{2N}}{\\pi^{2N-1}}\\sum_{k=0}^{\\infty}\\frac{(-1)^{k}}{(k+\\frac{1}{2})^{2N-1}\\Big(t^2+\\pi^2(k+\\frac{1}{2})^2\\Big)}, \\] and \\[\\mbox{sech}\\, t=\\sum_{j=0}^{N-1}\\frac{E_{2j}}{(2j)!}t^{2j}+\\Theta(t, N)\\frac{E_{2N}}{(2N)!}t^{2N} \\] with a suitable $0 < \\Theta(t, N) < 1$. Here $E_n$ are the Euler numbers. By using the obtained results, we deduce some inequalities and completely monotonic functions associated with the ratio of gamma functions. Furthermore, we give a (presumably new) quadratic recurrence relation for the Bernoulli numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-10T09:37:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B58",
      "26A48",
      "26D15"
    ],
    "keywords": [
      "euler numbers",
      "applications",
      "quadratic recurrence relation",
      "bernoulli numbers",
      "series representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160102192C"
    }
  }
}