{
  "id": "2109.15306",
  "version": "v4",
  "published": "2021-09-23T02:01:31.000Z",
  "updated": "2022-07-11T03:53:52.000Z",
  "title": "Lerch's $Φ$ and the Polylogarithm at the Negative Integers",
  "authors": [
    "Jose Risomar Sousa"
  ],
  "comment": "Fixed a few typos",
  "categories": [
    "math.NT"
  ],
  "abstract": "At the negative integers, there is a simple relation between the Lerch $\\Phi$ function and the polylogarithm. The literature has a formula for the polylogarithm at the negative integers, which utilizes the Stirling numbers of the second kind. Starting from that formula, we can deduce a simple closed formula for the Lerch $\\Phi$ function at the negative integers, where the Stirling numbers are not needed. Leveraging that finding, we also produce alternative formulae for the $k$-th derivatives of the cotangent and cosecant (ditto, tangent and secant), as simple functions of the negative polylogarithm and the Lerch $\\Phi$, respectively, which is evidence of the importance of these functions (they are less exotic than they seem). Lastly, we present a new formula for the Hurwitz zeta function at the positive integers using this novelty.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2022-07-11T03:53:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "negative integers",
      "polylogarithm",
      "hurwitz zeta function",
      "stirling numbers",
      "produce alternative formulae"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}