{
  "id": "1906.04311",
  "version": "v1",
  "published": "2019-06-10T22:38:16.000Z",
  "updated": "2019-06-10T22:38:16.000Z",
  "title": "Linear recurrences indexed by $\\mathbb{Z}$",
  "authors": [
    "Greg Muller"
  ],
  "comment": "25 pages, best viewed in color",
  "categories": [
    "math.CO",
    "math.RA"
  ],
  "abstract": "This note collects several general results about linear recurrences (also called linear difference equations) in unknowns indexed by the integers. We characterize a unique \\emph{reduced} system equivalent to a given linear recurrence, and construct a \\emph{solution matrix} which parametrizes the space of solutions to the original system. Several properties of solution matrices are shown, including a combinatorial characterization of bases and dimension of the space of solutions in terms of \\emph{juggling patterns}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-10T22:38:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear recurrence",
      "linear difference equations",
      "general results",
      "solution matrices",
      "note collects"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}