{
  "id": "math/0504525",
  "version": "v2",
  "published": "2005-04-26T01:21:18.000Z",
  "updated": "2005-11-02T14:22:10.000Z",
  "title": "A Telescoping method for Double Summations",
  "authors": [
    "William Y. C. Chen",
    "Qing-Hu Hou",
    "Yan-Ping Mu"
  ],
  "comment": "22 pages. to appear in J. Computational and Applied Mathematics",
  "categories": [
    "math.CO"
  ],
  "abstract": "We present a method to prove hypergeometric double summation identities. Given a hypergeometric term $F(n,i,j)$, we aim to find a difference operator $ L=a_0(n) N^0 + a_1(n) N^1 +...+a_r(n) N^r $ and rational functions $R_1(n,i,j),R_2(n,i,j)$ such that $ L F = \\Delta_i (R_1 F) + \\Delta_j (R_2 F)$. Based on simple divisibility considerations, we show that the denominators of $R_1$ and $R_2$ must possess certain factors which can be computed from $F(n, i,j)$. Using these factors as estimates, we may find the numerators of $R_1$ and $R_2$ by guessing the upper bounds of the degrees and solving systems of linear equations. Our method is valid for the Andrews-Paule identity, Carlitz's identities, the Ap\\'ery-Schmidt-Strehl identity, the Graham-Knuth-Patashnik identity, and the Petkov\\v{s}ek-Wilf-Zeilberger identity.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-11-02T14:22:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33F10",
      "68W30"
    ],
    "keywords": [
      "telescoping method",
      "hypergeometric double summation identities",
      "simple divisibility considerations",
      "hypergeometric term",
      "apery-schmidt-strehl identity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}