{
  "id": "1104.3778",
  "version": "v1",
  "published": "2011-04-19T15:33:59.000Z",
  "updated": "2011-04-19T15:33:59.000Z",
  "title": "The nearest neighbor recurrence coefficients for multiple orthogonal polynomials",
  "authors": [
    "Walter Van Assche"
  ],
  "comment": "22 pages",
  "journal": "J. Approx. Theory 163 (2011), 1427-1448",
  "doi": "10.1016/j.jat.2011.05.003",
  "categories": [
    "math.CA"
  ],
  "abstract": "We show that multiple orthogonal polynomials for r measures $(\\mu_1,...,\\mu_r)$ satisfy a system of linear recurrence relations only involving nearest neighbor multi-indices $\\vec{n}\\pm \\vec{e}_j$, where $\\vec{e}_j$ are the standard unit vectors. The recurrence coefficients are not arbitrary but satisfy a system of partial difference equations with boundary values given by the recurrence coefficients of the orthogonal polynomials with each of measures $\\mu_j$. We show how the Christoffel-Darboux formula for multiple orthogonal polynomials can be obtained easily using this information. We give explicit examples involving multiple Hermite, Charlier, Laguerre, and Jacobi polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-04-19T15:33:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C45",
      "42C05",
      "39A14"
    ],
    "keywords": [
      "multiple orthogonal polynomials",
      "nearest neighbor recurrence coefficients",
      "nearest neighbor multi-indices",
      "linear recurrence relations",
      "standard unit vectors"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1104.3778V"
    }
  }
}