{
  "id": "2101.06810",
  "version": "v1",
  "published": "2021-01-18T00:18:41.000Z",
  "updated": "2021-01-18T00:18:41.000Z",
  "title": "Classification of $K$-type formulas for the Heisenberg ultrahyperbolic operator $\\square_s$ for $\\widetilde{SL}(3,\\mathbb{R})$ and tridiagonal determinants for local Heun functions",
  "authors": [
    "Toshihisa Kubo",
    "Bent Ørsted"
  ],
  "comment": "67 pages",
  "categories": [
    "math.RT",
    "math.CA",
    "math.CO"
  ],
  "abstract": "The $K$-type formulas of the space of $K$-finite solutions to the Heisenberg ultrahyperbolic equation $\\square_sf=0$ for the non-linear group $\\widetilde{SL}(3,\\mathbb{R})$ are classified. This completes a previous study of Kable for the linear group $SL(m,\\mathbb{R})$ in the case of $m=3$, as well as generalizes our earlier results on a certain second order differential operator. As a by-product we also show several properties of certain sequences $\\{P_j(x;y)\\}_{j=0}^\\infty$ and $\\{Q_j(x;y)\\}_{j=0}^\\infty$ of tridiagonal determinants, whose generating functions are given by local Heun functions. In particular, it is shown that these sequences satisfy a certain arithmetic-combinatorial property, which we refer to as a palindromic property. We further show that classical sequences of Cayley continuants $\\{\\mathrm{Cay}_j(x;y)\\}_{j=0}^\\infty$ and Krawtchouk polynomials $\\{\\mathcal{K}_j(x;y)\\}_{j=0}^\\infty$ also admit this property. In the end a new proof of Sylvester's formula for certain tridiagonal determinant $\\mathrm{Sylv}(x;n)$ is provided from a representation theory point of view.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-18T00:18:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E46",
      "17B10",
      "05B20",
      "33C05",
      "33C45",
      "33E30"
    ],
    "keywords": [
      "local heun functions",
      "heisenberg ultrahyperbolic operator",
      "tridiagonal determinant",
      "type formulas",
      "second order differential operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 67,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}