{
  "id": "math/0607514",
  "version": "v1",
  "published": "2006-07-20T22:20:09.000Z",
  "updated": "2006-07-20T22:20:09.000Z",
  "title": "On asymptotics, Stirling numbers, Gamma function and polylogs",
  "authors": [
    "Daniel B. Grünberg"
  ],
  "comment": "24 pages, to appear in Results for Mathematics",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "We apply the Euler--Maclaurin formula to find the asymptotic expansion of the sums $\\sum_{k=1}^n (\\log k)^p / k^q$, ~$\\sum k^q (\\log k)^p$, ~$\\sum (\\log k)^p /(n-k)^q$, ~$\\sum 1/k^q (\\log k)^p $ in closed form to arbitrary order ($p,q \\in\\N$). The expressions often simplify considerably and the coefficients are recognizable constants. The constant terms of the asymptotics are either $\\zeta^{(p)}(\\pm q)$ (first two sums), 0 (third sum) or yield novel mathematical constants (fourth sum). This allows numerical computation of $\\zeta^{(p)}(\\pm q)$ faster than any current software. One of the constants also appears in the expansion of the function $\\sum_{n\\geq 2} (n\\log n)^{-s}$ around the singularity at $s=1$; this requires the asymptotics of the incomplete gamma function. The manipulations involve polylogs for which we find a representation in terms of Nielsen integrals, as well as mysterious conjectures for Bernoulli numbers. Applications include the determination of the asymptotic growth of the Taylor coefficients of $(-z/\\log(1-z))^k$. We also give the asymptotics of Stirling numbers of first kind and their formula in terms of harmonic numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-07-20T22:20:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A10",
      "11A07",
      "30B10"
    ],
    "keywords": [
      "stirling numbers",
      "incomplete gamma function",
      "yield novel mathematical constants",
      "taylor coefficients",
      "asymptotic growth"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......7514G"
    }
  }
}