{
  "id": "1703.06401",
  "version": "v1",
  "published": "2017-03-19T08:44:54.000Z",
  "updated": "2017-03-19T08:44:54.000Z",
  "title": "On some combinatorial identities and harmonic sums",
  "authors": [
    "Necdet Batir"
  ],
  "comment": "to appear in Int. J. Number Theory, 2017",
  "doi": "10.1142/S179304211750097X",
  "categories": [
    "math.NT"
  ],
  "abstract": "For any $m,n\\in\\mathbb{N}$ we first give new proofs for the following well known combinatorial identities \\begin{equation*} S_n(m)=\\sum\\limits_{k=1}^n\\binom{n}{k}\\frac{(-1)^{k-1}}{k^m}=\\sum\\limits_{n\\geq r_1\\geq r_2\\geq...\\geq r_m\\geq 1}\\frac{1}{r_1r_2\\cdots r_m} \\end{equation*} and $$ \\sum\\limits_{k=1}^n(-1)^{n-k}\\binom{n}{k}k^n = n!, $$ and then we produce the generating function and an integral representation for $S_n(m)$. Using them we evaluate many interesting finite and infinite harmonic sums in closed form. For example, we show that $$ \\zeta(3)=\\frac{1}{9}\\sum\\limits_{n=1}^\\infty\\frac{H_n^3+3H_nH_n^{(2)}+2H_n^{(3)}}{2^n}, $$ and $$ \\zeta(5)=\\frac{2}{45}\\sum\\limits_{n=1}^{\\infty}\\frac{H_n^4+6H_n^2H_n^{(2)}+8H_nH_n^{(3)}+3\\left(H_n^{(2)}\\right)^2+6H_n^{(4)}}{n2^n}, $$ where $H_n^{(i)}$ are generalized harmonic numbers defined below.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-19T08:44:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A10"
    ],
    "keywords": [
      "combinatorial identities",
      "infinite harmonic sums",
      "generalized harmonic numbers",
      "integral representation",
      "interesting finite"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "World Scientific"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}