{
  "id": "0708.0563",
  "version": "v2",
  "published": "2007-08-03T18:50:33.000Z",
  "updated": "2008-02-26T14:53:48.000Z",
  "title": "Probabilistic implications of symmetries of q-Hermite and Al-Salam-Chihara polynomials",
  "authors": [
    "Paweł J. Szabłowski"
  ],
  "comment": "7 pages",
  "journal": "Infin. Dimens. Anal. Quantum Probab. Relat. Top. Vol. 11, No. 4 (2008) 513-522",
  "categories": [
    "math.PR",
    "math.CV"
  ],
  "abstract": "We prove the existence of stationary random fields with linear regressions for $q>1$ and thus close an open question posed by W. Bryc et al.. We prove this result by describing a discrete 1 dimensional conditional distribution and then checking Chapman-Kolmogorov equation. Support of this distribution consist of zeros of certain Al-Salam-Chihara polynomials. To find them we refer to and expose known result concerning addition of $q-$ exponential function. This leads to generalization of a well known formula $(x+y)^{n}% =\\sum_{i=0}^{n}\\binom{n}{k}i^{k}H_{n-k}(x) H_{k}(-iy) ,$ where $H_{k}(x) $ denotes $k-$th Hermite polynomial.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-02-26T14:53:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J27",
      "33C45"
    ],
    "keywords": [
      "al-salam-chihara polynomials",
      "probabilistic implications",
      "symmetries",
      "dimensional conditional distribution",
      "stationary random fields"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0708.0563S"
    }
  }
}