{
  "id": "0705.1812",
  "version": "v3",
  "published": "2007-05-13T02:27:58.000Z",
  "updated": "2007-08-21T06:49:48.000Z",
  "title": "The Cauchy Operator for Basic Hypergeometric Series",
  "authors": [
    "Vincent Y. B. Chen",
    "Nancy S. S. Gu"
  ],
  "comment": "21 pages, to appear in Advances in Applied Mathematics",
  "categories": [
    "math.CO"
  ],
  "abstract": "We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine's ${}_2\\phi_1$ transformation formula and Sears' ${}_3\\phi_2$ transformation formula can be easily obtained by the symmetric property of some parameters in operator identities. The Cauchy operator involves two parameters, and it can be considered as a generalization of the operator $T(bD_q)$. Using this operator, we obtain extensions of the Askey-Wilson integral, the Askey-Roy integral, Sears' two-term summation formula, as well as the $q$-analogues of Barnes' lemmas. Finally, we find that the Cauchy operator is also suitable for the study of the bivariate Rogers-Szeg\\\"o polynomials, or the continuous big $q$-Hermite polynomials.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2007-08-21T06:49:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A30",
      "33D05",
      "33D15"
    ],
    "keywords": [
      "basic hypergeometric series",
      "cauchy operator",
      "transformation formula",
      "cauchy augmentation operator",
      "two-term summation formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0705.1812C"
    }
  }
}