{
  "id": "1307.0599",
  "version": "v2",
  "published": "2013-07-02T07:09:13.000Z",
  "updated": "2014-09-20T06:26:36.000Z",
  "title": "New steps in walks with small steps in the quarter plane",
  "authors": [
    "Irina Kurkova",
    "Kilian Raschel"
  ],
  "comment": "47 pages, 8 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this article we obtain new expressions for the generating functions counting (non-singular) walks with small steps in the quarter plane. Those are given in terms of infinite series, while in the literature, the standard expressions use solutions to boundary value problems. We illustrate our results with three examples (an algebraic case, a transcendental D-finite case, and an infinite group model).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-07-02T07:09:13.000Z",
      "abstract": "In this article we present a new approach for finding the generating function counting (non-singular) walks with small steps in the quarter plane. We obtain its explicit expression in terms of infinite series, while in the literature, its standard expression is given in terms of solutions to boundary value problems. As an illustration of the computational interest of our approach, we work out these new formulae for Kreweras' model: this leads to an expression for the generating function with Weierstrass Zeta functions, and eventually (using transformation theory of elliptic functions) to the well-known algebraic expression.",
      "comment": "36 pages, 6 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-09-20T06:26:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "30F10",
      "30D05"
    ],
    "keywords": [
      "small steps",
      "quarter plane",
      "generating function",
      "weierstrass zeta functions",
      "boundary value problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.0599K"
    }
  }
}