{
  "id": "2409.18609",
  "version": "v1",
  "published": "2024-09-27T10:12:33.000Z",
  "updated": "2024-09-27T10:12:33.000Z",
  "title": "Hankel Determinants for a Class of Weighted Lattice Paths",
  "authors": [
    "Ying Wang",
    "Zihao Zhang"
  ],
  "comment": "14 pages; 0 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, our primary goal is to calculate the Hankel determinants for a class of lattice paths, which are distinguished by the step set consisting of \\(\\{(1,0), (2,0), (k-1,1), (-1,1)\\}\\), where the parameter \\(k\\geq 4\\). These paths are constrained to return to the $x$-axis and remain above the \\(x\\)-axis. When calculating for \\(k = 4\\), the problem essentially reduces to determining the Hankel determinant of \\(E(x)\\), where \\(E(x)\\) is defined as \\[ E(x) = \\frac{a}{E(x)x^2(dx^2 - bx - 1) + cx^2 + bx + 1}. \\] Our approach involves employing the Sulanke-Xin continued fraction transform to derive a set of recurrence relations, which in turn yield the desired results. For \\(k \\geq 5\\), we utilize a class of shifted periodic continued fractions as defined by Wang-Xin-Zhai, thereby obtaining the results presented in this paper.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-27T10:12:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A15",
      "05A15",
      "11B83"
    ],
    "keywords": [
      "hankel determinant",
      "weighted lattice paths",
      "sulanke-xin continued fraction transform",
      "step set",
      "shifted periodic continued fractions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}