{
  "id": "math/0501441",
  "version": "v3",
  "published": "2005-01-25T12:02:28.000Z",
  "updated": "2005-08-31T09:38:57.000Z",
  "title": "A q-Analogue of Faulhaber's Formula for Sums of Powers",
  "authors": [
    "Victor J. W. Guo",
    "Jiang Zeng"
  ],
  "comment": "19 pages",
  "journal": "Electron. J. Combin. 11(2) (2005), #R19",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $$ S_{m,n}(q):=\\sum_{k=1}^{n}\\frac{1-q^{2k}}{1-q^2} (\\frac{1-q^k}{1-q})^{m-1}q^{\\frac{m+1}{2}(n-k)}. $$ Generalizing the formulas of Warnaar and Schlosser, we prove that there exist polynomials $P_{m,k}(q)\\in\\mathbb{Z}[q]$ such that $$ S_{2m+1,n}(q) =\\sum_{k=0}^{m}(-1)^kP_{m,k}(q) \\frac{(1-q^n)^{m+1-k}(1-q^{n+1})^{m+1-k}q^{kn}} {(1-q^2)(1-q)^{2m-3k}\\prod_{i=0}^{k}(1-q^{m+1-i})}, $$ and solve a problem raised by Schlosser. We also show that there is a similar formula for the following $q$-analogue of alternating sums of powers: $$ T_{m,n}(q):=\\sum_{k=1}^{n}(-1)^{n-k} (\\frac{1-q^k}{1-q})^{m}q^{\\frac{m}{2}(n-k)}. $$",
  "revisions": [
    {
      "version": "v3",
      "updated": "2005-08-31T09:38:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A30",
      "05A15"
    ],
    "keywords": [
      "faulhabers formula",
      "q-analogue",
      "similar formula"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......1441G"
    }
  }
}