{
  "abstract": "Let $\\mathbb{Z}^-=\\{-1,-2,\\dotsc\\}$ and $\\mathbb{N}=\\{1,2,\\dotsc\\}$. In 2023, Qi and Lim gave two claims for summing the infinite series $\\sum_{k=1}^{\\infty}\\binom{2k}{k}\\frac{1}{α+k}\\bigl(\\frac{\\pm1}{4}\\bigr)^k$ for $α\\in\\mathbb{C}\\setminus\\mathbb{Z}^-$. In this paper, we confirm these claims by expressing the infinite series $\\sum_{k=1}^{\\infty}\\binom{2k}{k}\\frac{1}{α+k}\\bigl(\\frac{z}{4}\\bigr)^k$ and the finite series $\\sum_{k=1}^{n}\\binom{2k}{k}\\frac{1}{α+k}\\bigl(\\frac{z}{4}\\bigr)^k$ in terms of the Gauss hypergeometric function ${}_2F_1\\bigl[\\tfrac12,α;1+α;z\\bigr]$ and a generalized hypergeometric function ${}_3F_2$, for $|z|\\le1$ and $n\\in\\mathbb{N}$. Using the Euler integral representation, we derive closed forms, in terms of finite sums of sines and cosines of $\\arcsin\\sqrt{z}\\,$, for ${}_2F_1\\bigl[\\tfrac12,\\tfrac12+n;\\tfrac32+n;z\\bigr]$, ${}_2F_1\\bigl[\\tfrac12,1+n;2+n;z\\bigr]$, two related ${}_3F_2$ functions, and the incomplete beta functions $B_z\\bigl(\\frac12+n,\\frac12\\bigr)$ and $B_z\\bigl(1+n,\\frac12\\bigr)$. By the Pfaff transformation, we obtain closed forms of several further Gauss hypergeometric functions with parameters shifted by $n$. We also give a closed form of $\\bigl[(1-z)\\frac{\\operatorname{d}}{\\operatorname{d}z}(1-z)\\bigr]^n\\frac{\\arcsin\\sqrt{z}\\,}{\\sqrt{z(1-z)}\\,}$ and a combinatorial identity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.05193",
  "author_total": 4,
  "authors": [
    {
      "name": "Ganesh Bahadur Basnet"
    },
    {
      "name": "Narayan Prasad Pahari"
    },
    {
      "name": "Feng Qi"
    },
    {
      "name": "Arjun Kumar Rathie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/75hr98r8hzpqf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.05193",
  "fulltext_url": null,
  "id": "75hr98r8hzpqf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.05193",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GM"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7201868562",
  "pdf_url": "https://arxiv.org/pdf/2608.05193",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/75hr98r8hzpqf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2608.05193",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.05193"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Advanced Mathematical Identities",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Infinite and finite series involving central binomial coefficients and closed forms of generalized hypergeometric functions",
  "title_text": "Infinite and finite series involving central binomial coefficients and closed forms of generalized hypergeometric functions",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/75hr98r8hzpqf",
  "venue": "arXiv",
  "versions": []
}