{
  "id": "1806.05515",
  "version": "v1",
  "published": "2018-06-14T12:59:22.000Z",
  "updated": "2018-06-14T12:59:22.000Z",
  "title": "On poly-Euler numbers of the second kind",
  "authors": [
    "Takao Komatsu"
  ],
  "comment": "This manuscript has been accepted for publication in Bessatsu of Algebraic Number Theory and Related Topics 2016",
  "categories": [
    "math.NT"
  ],
  "abstract": "For an integer $k$, define poly-Euler numbers of the second kind $\\widehat E_n^{(k)}$ ($n=0,1,\\dots$) by $$ \\frac{{\\rm Li}_k(1-e^{-4 t})}{4\\sinh t}=\\sum_{n=0}^\\infty\\widehat E_n^{(k)}\\frac{t^n}{n!}\\,. $$ When $k=1$, $\\widehat E_n=\\widehat E_n^{(1)}$ are {\\it Euler numbers of the second kind} or {\\it complimentary Euler numbers} defined by $$ \\frac{t}{\\sinh t}=\\sum_{n=0}^\\infty\\widehat E_n\\frac{t^n}{n!}\\,. $$ Euler numbers of the second kind were introduced as special cases of hypergeometric Euler numbers of the second kind in \\cite{KZ}, so that they would supplement hypergeometric Euler numbers. In this paper, we give several properties of Euler numbers of the second kind. In particular, we determine their denominators. We also show several properties of poly-Euler numbers of the second kind, including duality formulae and congruence relations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-14T12:59:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B68",
      "05A15",
      "11M41"
    ],
    "keywords": [
      "second kind",
      "supplement hypergeometric euler numbers",
      "define poly-euler numbers",
      "complimentary euler numbers",
      "congruence relations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}