{
  "id": "1710.07127",
  "version": "v1",
  "published": "2017-10-19T13:18:52.000Z",
  "updated": "2017-10-19T13:18:52.000Z",
  "title": "Identities involving Bernoulli and Euler polynomials",
  "authors": [
    "Horst Alzer",
    "Semyon Yakubovich"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "We present various identities involving the classical Bernoulli and Euler polynomials. Among others, we prove that $$ \\sum_{k=0}^{[n/4]}(-1)^k {n\\choose 4k}\\frac{B_{n-4k}(z) }{2^{6k}} =\\frac{1}{2^{n+1}}\\sum_{k=0}^{n} (-1)^k \\frac{1+i^k}{(1+i)^k} {n\\choose k}{B_{n-k}(2z)} $$ and $$ \\sum_{k=1}^{n} 2^{2k-1} {2n\\choose 2k-1} B_{2k-1}(z) = \\sum_{k=1}^n k \\, 2^{2k} {2n\\choose 2k} E_{2k-1}(z). $$ Applications of our results lead to formulas for Bernoulli and Euler numbers, like, for instance, $$ n E_{n-1} =\\sum_{k=1}^{[n/2]} \\frac{2^{2k}-1}{k} (2^{2k}-2^n){n\\choose 2k-1} B_{2k}B_{n-2k}. $$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-19T13:18:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B68",
      "11M06",
      "12E10"
    ],
    "keywords": [
      "euler polynomials",
      "identities",
      "euler numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}