{
  "abstract": "The large-degree phase of the Jacobi polynomial $P_n^{(α,β)}(\\cosθ)$ contains, at each order in $\\Nn^{-1}$, $\\Nn=n+(α+β+1)/2$, an additive constant $κ_r$ that the phase equation does not determine. These constants are antisymmetric in $(α^2,β^2)$ and vanish for $α=β$. We obtain three results. First, on the diagonal $α=β$ the logarithm of the midpoint amplitude is computed in closed form: if $L_n$ denotes the coefficient of $(2\\Nn)^{-n}$ in that logarithm, then $Υ_A(t)=\\sum_{n\\ge1}L_nt^n/(n-1)!$ equals \\[ Υ_A(t)=\\frac{e^{t}\\bigl(\\cosh2αt-\\cosh t\\bigr)}{\\sinh 2t}. \\] The proof is algebraic: the identity is established for half-odd-integer $α$, where the expansion terminates, by means of two classical recurrences for Bessel polynomials, and is extended to all $α$ by a degree argument. Second, the derivative of the logarithm of the amplitude with respect to $α^2$ at the Legendre point $α=β=0$ is computed in closed form to all orders. An exact reduction of an Appell $F_3$ function to a Gauss function shows that it is a parameter derivative of a zero-balanced ${}_2F_1$ at argument $2$; the real part of its expansion is a trigamma series, and the imaginary part is the unique formal solution of an explicit central-difference equation. Third, we deduce that the coefficients $w_m$ of an explicit even power series attached to the imaginary part satisfy $ν_2(w_m)=-2ν_2((2m)!)$ for every $m\\ge0$ ($ν_2$ the $2$-adic valuation). This determines, for every $r$, the exact power of $2$ in the denominator of the coefficient of $α^2$ in the polynomial that represents $κ_r$. We also give a factorisation of the amplitude into products of pairs of Gauss functions, further valuation formulas, and a numerical illustration for the zeros.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29531",
  "author_total": 1,
  "authors": [
    {
      "name": "Iván Area"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ekhhnn53qxv7r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29531",
  "fulltext_url": null,
  "id": "ekhhnn53qxv7r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29531",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7214400205",
  "pdf_url": "https://arxiv.org/pdf/2609.29531",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ekhhnn53qxv7r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2609.29531",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29531"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The ultraspherical background of the Jacobi phase expansion, in closed form",
  "title_text": "The ultraspherical background of the Jacobi phase expansion, in closed form",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/ekhhnn53qxv7r",
  "venue": "arXiv",
  "versions": []
}