{
  "id": "0910.1746",
  "version": "v1",
  "published": "2009-10-09T13:50:10.000Z",
  "updated": "2009-10-09T13:50:10.000Z",
  "title": "An Operator Approach to the Al-Salam-Carlitz Polynomials",
  "authors": [
    "William Y. C. Chen",
    "Husam L. Saad",
    "Lisa H. Sun"
  ],
  "comment": "15 pages",
  "categories": [
    "math.CA",
    "math.CO"
  ],
  "abstract": "We present an operator approach to Rogers-type formulas and Mehler's formulas for the Al-Salam-Carlitz polynomials $U_n(x,y,a;q)$. By using the q-exponential operator, we obtain a Rogers-type formula which leads to a linearization formula. With the aid of a bivariate augmentation operator, we get a simple derivation of Mehler's formula due to by Al-Salam and Carlitz, which requires a terminating condition on a ${}_3\\phi_2$ series. By means of the Cauchy companion augmentation operator, we obtain Mehler's formula in a similar form, but it does not need the terminating condition. We also give several identities on the generating functions for products of the Al-Salam-Carlitz polynomials which are extensions of formulas for Rogers-Szeg\\\"o polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-10-09T13:50:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "02.30.-f",
      "02.10.De"
    ],
    "keywords": [
      "al-salam-carlitz polynomials",
      "operator approach",
      "mehlers formula",
      "rogers-type formula",
      "cauchy companion augmentation operator"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1063/1.3321603",
      "journal": "Journal of Mathematical Physics",
      "year": 2010,
      "month": "Apr",
      "volume": 51,
      "number": 4,
      "pages": 3502
    },
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010JMP....51d3502C"
    }
  }
}