{
  "id": "1012.4774",
  "version": "v1",
  "published": "2010-12-21T19:33:07.000Z",
  "updated": "2010-12-21T19:33:07.000Z",
  "title": "On convolutions of Euler numbers",
  "authors": [
    "Zhi-Wei Sun"
  ],
  "comment": "6 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "We show that if p is an odd prime then $$\\sum_{k=0}^{p-1}E_kE_{p-1-k}=1 (mod p)$$ and $$\\sum_{k=0}^{p-3}E_kE_{p-3-k}=(-1)^{(p-1)/2}2E_{p-3} (mod p),$$ where E_0,E_1,E_2,... are Euler numbers. Moreover, we prove that for any positive integer n and prime number p>2n+1 we have $$\\sum_{k=0}^{p-1+2n}E_kE_{p-1+2n-k}=s(n) (mod p)$$ where s(n) is an integer only depending on n.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-12-21T19:33:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B68",
      "11A07",
      "05A99"
    ],
    "keywords": [
      "euler numbers",
      "convolutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.4774S"
    }
  }
}