{
  "id": "math/0612430",
  "version": "v2",
  "published": "2006-12-15T08:57:43.000Z",
  "updated": "2007-05-11T03:51:39.000Z",
  "title": "The Bivariate Rogers-Szegö Polynomials",
  "authors": [
    "William Y. C. Chen",
    "Husam L. Saad",
    "Lisa H. Sun"
  ],
  "comment": "16 pages, revised version, to appear in J. Phys. A: Math. Theor",
  "categories": [
    "math.CO"
  ],
  "abstract": "We present an operator approach to deriving Mehler's formula and the Rogers formula for the bivariate Rogers-Szeg\\\"{o} polynomials $h_n(x,y|q)$. The proof of Mehler's formula can be considered as a new approach to the nonsymmetric Poisson kernel formula for the continuous big $q$-Hermite polynomials $H_n(x;a|q)$ due to Askey, Rahman and Suslov. Mehler's formula for $h_n(x,y|q)$ involves a ${}_3\\phi_2$ sum and the Rogers formula involves a ${}_2\\phi_1$ sum. The proofs of these results are based on parameter augmentation with respect to the $q$-exponential operator and the homogeneous $q$-shift operator in two variables. By extending recent results on the Rogers-Szeg\\\"{o} polynomials $h_n(x|q)$ due to Hou, Lascoux and Mu, we obtain another Rogers-type formula for $h_n(x,y|q)$. Finally, we give a change of base formula for $H_n(x;a|q)$ which can be used to evaluate some integrals by using the Askey-Wilson integral.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-05-11T03:51:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A30",
      "33D45"
    ],
    "keywords": [
      "polynomials",
      "rogers formula",
      "nonsymmetric poisson kernel formula",
      "askey-wilson integral",
      "base formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}