{
  "id": "2310.12697",
  "version": "v1",
  "published": "2023-10-19T12:51:58.000Z",
  "updated": "2023-10-19T12:51:58.000Z",
  "title": "Closed-form formulas, determinantal expressions, recursive relations, power series, and special values of several functions used in Clark--Ismail's two conjectures",
  "authors": [
    "Yan-Fang Li",
    "Dongkyu Lim",
    "Feng Qi"
  ],
  "comment": "23 pages",
  "categories": [
    "math.CA",
    "math.CO",
    "math.NT"
  ],
  "abstract": "In the paper, by virtue of the famous formula of Fa\\`a di Bruno, with the aid of several identities of partial Bell polynomials, by means of a formula for derivatives of the ratio of two differentiable functions, and with availability of other techniques, the authors establish closed-form formulas in terms of the Bernoulli numbers and the second kind Stirling numbers, present determinantal expressions, derive recursive relations, obtain power series, and compute special values of the function $\\frac{v^j}{1-\\operatorname{e}^{-v}}$, its derivatives, and related ones used in Clark--Ismail's two conjectures. By these results, the authors also discover a formula for the determinant of a Hessenberg matrix and derive logarithmic convexity of a sequence related to the function and its derivatives.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-19T12:51:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33B10",
      "15A15",
      "26A24",
      "26A48",
      "26A51",
      "33B15",
      "44A10",
      "41A58"
    ],
    "keywords": [
      "power series",
      "special values",
      "determinantal expressions",
      "recursive relations",
      "clark-ismails"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}