{
  "id": "1811.03285",
  "version": "v1",
  "published": "2018-11-08T06:11:41.000Z",
  "updated": "2018-11-08T06:11:41.000Z",
  "title": "Combinatorial expressions for the tau functions of $q$-Painlevé V and III equations",
  "authors": [
    "Yuya Matsuhira",
    "Hajime Nagoya"
  ],
  "comment": "13 pages",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We derive series representations for the tau functions of the $q$-Painlev\\'e V, $\\mathrm{III_1}$, $\\mathrm{III_2}$, and $\\mathrm{III_3}$ equations, as degenerations of the tau functions of the $q$-Painlev\\'e VI equation in \\cite{JNS}. Our tau functions are expressed in terms of $q$-Nekrasov functions. Thus, our series representations for the tau functions have explicit combinatorial structures. We show that general solutions to the $q$-Painlev\\'e V, $\\mathrm{III_1}$, $\\mathrm{III_2}$, and $\\mathrm{III_3}$ equations are written by our tau functions. We also prove that our tau functions for the $q$-Painlev\\'e $\\mathrm{III_1}$, $\\mathrm{III_2}$, and $\\mathrm{III_3}$ equations satisfy the three-term bilinear equations for them.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-08T06:11:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tau functions",
      "combinatorial expressions",
      "painleve vi equation",
      "three-term bilinear equations",
      "explicit combinatorial structures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}