{
  "id": "1411.3527",
  "version": "v1",
  "published": "2014-11-13T12:49:48.000Z",
  "updated": "2014-11-13T12:49:48.000Z",
  "title": "Finite-dimensional representations of difference operators, and the identification of remarkable matrices",
  "authors": [
    "Francesco Calogero"
  ],
  "comment": "29 pages",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "Two square matrices of (arbitrary) order N are introduced. They are defined in terms of N arbitrary numbers z_{n}, and of an arbitrary additional parameter (a respectively q), and provide finite-dimensional representations of the two operators acting on a function f(z) as follows: [f(z+a)-f(z)]/a respectively [f(qz)-f(z)]/[(q-1)z]. These representations are exact---in a sense explained in the paper---when the function f(z) is a polynomial in z of degree less than N. This formalism allows to transform difference equations valid in the space of polynomials of degree less than N into corresponding matrix-vector equations. As an application of this technique several remarkable square matrices of order N are identified, which feature explicitly N arbitrary numbers z_{n}, or the N zeros of polynomials belonging to the Askey and q-Askey schemes. Several of these findings have a Diophantine character.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-13T12:49:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite-dimensional representations",
      "difference operators",
      "identification",
      "transform difference equations valid",
      "arbitrary numbers"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1063/1.4915291",
      "journal": "Journal of Mathematical Physics",
      "year": 2015,
      "month": "Mar",
      "volume": 56,
      "number": 3,
      "pages": "033506"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015JMP....56c3506C"
    }
  }
}