{
  "abstract": "In the present paper, with the help of the Faà di Bruno formula and several identities of partial Bell polynomials, the author establishes explicit formulas of the Gauss hypergeometric functions \\begin{gather*} {\\,}_2F_1\\biggl(\\frac{1-n}{2},\\frac{2-n}{2};\\frac{3}{2}-m;z^2\\biggr), \\quad {\\,}_2F_1\\biggl(-\\frac{n}{2},\\frac{1-n}{2};\\frac{1}{2}-m;z^2\\biggr),\\\\ {\\,}_2F_1\\biggl(a,a+\\frac{1}{2};\\frac{3}{2}-m;z^2\\biggr), \\quad {\\,}_2F_1\\biggl(a,a+\\frac{1}{2};\\frac{1}{2}-m;z^2\\biggr) \\end{gather*} for $m,n\\in\\mathbb{N}$ and $a\\in\\mathbb{C}$, and then derives two combinatorial identities \\begin{equation*} \\sum_{k=0}^{m}\\frac{2^k}{k!} \\binom{2m-2k}{m-k} \\sum_{\\ell=0}^{k} \\frac{(-1)^\\ell}{2^\\ell} \\frac{(2k-2\\ell-1)!!}{(n-\\ell)!} \\binom{2k-\\ell-1}{\\ell-1} =\\frac{1}{n!}\\binom{2m-n}{m} \\end{equation*} and \\begin{equation*} \\sum_{k=1}^{m}\\frac{1}{(k!)^2}\\binom{2m-2k}{m-k} \\sum_{\\ell=1}^{k} \\binom{k}{\\ell}\\ell(2k-\\ell-1)! (2a)_\\ell =\\binom{2m+2a}{m}-\\binom{2m}{m}, \\end{equation*} where $m\\in\\mathbb{N}_0$, $n\\in\\mathbb{Z}$, and $a\\in\\mathbb{C}$. These newly-established identities extend and generalize the beautiful combinatorial identity \\begin{equation*} \\sum_{k=0}^{n} \\frac{2^{k}}{k!}\\binom{2n-2k}{n-k} \\sum_{j=0}^{k}\\frac{(-1)^{j}}{2^j} \\frac{(2k-2j-1)!!}{(n-j)!} \\binom{2k-j-1}{j-1} =\\frac{1}{n!}, \\quad n\\in\\mathbb{N}_0, \\end{equation*} which was obtained in Theorem 4 of the paper \"F. Qi, C.-Y. He, and D. Lim, Explicit formulas of two Gauss hypergeometric functions and several combinatorial identities, Discrete Appl. Math. Vol. 393 (2026), 215--229. DOI: https://doi.org/10.1016/j.dam.2026.06.023.\"",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.10643",
  "author_total": 1,
  "authors": [
    {
      "name": "Feng Qi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cggh7vax7mcb1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.10643",
  "fulltext_url": null,
  "id": "cggh7vax7mcb1",
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  "landing_url": "https://arxiv.org/abs/2607.10643",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
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      "arxiv",
      "math.NT"
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      "openalex",
      "2602|Advanced Mathematical Identities"
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  ],
  "openalex": "W7168238275",
  "pdf_url": "https://arxiv.org/pdf/2607.10643",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cggh7vax7mcb1.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2607.10643",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.10643"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Combinatorial identities derived from explicit formulas of Gauss hypergeometric functions",
  "title_text": "Combinatorial identities derived from explicit formulas of Gauss hypergeometric functions",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:51Z",
  "url": "https://research.draconitic.com/paper/cggh7vax7mcb1",
  "venue": "arXiv",
  "versions": []
}