{
  "id": "2207.01935",
  "version": "v1",
  "published": "2022-07-05T10:18:28.000Z",
  "updated": "2022-07-05T10:18:28.000Z",
  "title": "Another Approach on Power Sums",
  "authors": [
    "Christoph Muschielok"
  ],
  "comment": "8 pages, 0 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "We show that explicit forms for certain polynomials~$\\psi^{(a)}_m(n)$ with the property \\[ \\psi^{(a+1)}_m(n) = \\sum_{\\nu=1}^n \\psi_m^{(a)}(\\nu) \\] can be found (here, $a,m,n\\in\\mathbb{N}_0$). We use these polynomials as a basis to express the monomials~$n^m$. Once the expansion coefficients are determined, we can express the $m$-th power sums~$S^{(a)}_m(n)$ of any order $a$, \\[ S^{(a)}_m(n) = \\sum_{\\nu_a = 1}^n \\cdots \\sum_{\\nu_2 = 1}^{\\nu_3} \\sum_{\\nu_1=1}^{\\nu_2} \\nu_1^m, \\] in a very convenient way by exploiting the summation property of the $\\psi_m^{(a)}$, \\[ S^{(a)}_m(n) = \\sum_k c_{mk} \\psi_k^{(a)}(n). \\]",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-05T10:18:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A10"
    ],
    "keywords": [
      "power sums",
      "convenient way",
      "explicit forms",
      "th power",
      "expansion coefficients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}