{
  "annotation_id": "121044ec-d502-4072-a006-942b6260cf37",
  "date_created": "2026-02-17T05:53:19.852000Z",
  "date_modified": "2026-02-17T05:53:19.852000Z",
  "file_hash": "c0763a17fe13b3487f3cbd2e1ba1ee530c12e25cd34b9e7c6803861002faff72",
  "private": false,
  "record": {
    "abstract": "In this paper, we present and prove that the coefficients $u_n$ and $v_n$ in the series expansions $F^2(a,b;c;z) = \\sum_{n=0}^\\infty u_n z^n$ and $F^3(a,b;c;z) = \\sum_{n=0}^\\infty v_n z^n$ ($a,b,c,z \\in \\mathbb{C}$ and $-c \\notin \\mathbb{N} \\cup \\{0\\}$) satisfy second- and third-order linear recurrence relations, respectively, where $F(a,b;c;x)$ denotes the Gaussian hypergeometric function and $\\mathbb{C}$ is the complex plane. Our results provide recurrence relations for the Maclaurin coefficients of the squares and cubes of several classical special functions in the complex domain, including zero-balanced Gauss hypergeometric functions, elliptic integrals, as well as classical orthogonal polynomials such as Chebyshev, Legendre, Gegenbauer, and Jacobi polynomials. As applications, we first establish the monotonicity of a function involving Gauss hypergeometric functions and then present a new proof of the well-known Clausen\u0027s formula.",
    "arxiv_id": "2601.09154",
    "authors": [
      "Zhong-Xuan Mao",
      "Jing-Feng Tian"
    ],
    "categories": [
      "math.CA",
      "math.CV"
    ],
    "license": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
    "title": "Recurrence relations and applications for the Maclaurin coefficients of squared and cubic hypergeometric functions",
    "url": "https://arxiv.org/abs/2601.09154",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "5c8baa86-2e17-48c2-ab1b-09d034a7a85a",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}