{
  "id": "1402.0773",
  "version": "v2",
  "published": "2014-02-04T15:44:24.000Z",
  "updated": "2014-07-01T13:22:37.000Z",
  "title": "On linearly related sequences of difference derivatives of discrete orthogonal polynomials",
  "authors": [
    "R. Alvarez-Nodarse",
    "J. Petronilho",
    "N. C. Pinzon-Cortes",
    "R. Sevinik-Adiguzel"
  ],
  "comment": "20 pages, Journal of Computational and Applied (2014) Journal of Computational and Applied Mathematics",
  "doi": "10.1016/j.cam.2014.06.018",
  "categories": [
    "math.CA"
  ],
  "abstract": "Let D_v the difference operator and q-difference operators defined by D_\\omega p(x) = \\frac{p(x+\\omega)-p(x)}{\\omega} and D_q p(x) = \\frac{p(qx)-p(x)}{(q-1)x}, respectively. Let U and V be two moment regular linear functionals and let (P_n)_n and Q_n)_n be their corresponding orthogonal polynomial sequences (OPS). We discuss an inverse problem in the theory of discrete orthogonal polynomials involving the above two OPS assuming that their difference derivatives $D_\\nu$ of higher orders m and k (resp.) are connected by a linear algebraic structure relation such as $$ \\sum_{i=0}^M a_{i,n} D_\\nu^m P_{n+m-i}(x) = \\sum_{i=0}^N b_{i,n} D_\\nu^k Q_{n+k-i}(x), \\quad n\\geq 0, $$ where $M,N,m,k=0,1,2,... Under certain conditions, we prove that U and V are related by a rational factor \\c{c} (in the distributional sense). Moreover, when m\\neq k then both U and V are D_v-semiclassical functionals. This leads us to the concept of (M,N)-D_v-coherent pair of order (m,k) extending to the discrete case several previous works. As an application we consider the OPS with respect to a certain following Sobolev-type discrete inner product.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-07-01T13:22:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C45",
      "41A10",
      "42C05"
    ],
    "keywords": [
      "discrete orthogonal polynomials",
      "difference derivatives",
      "linearly related sequences",
      "sobolev-type discrete inner product",
      "moment regular linear functionals"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.0773A"
    }
  }
}