{
  "abstract": "A formal large-degree phase expansion associated with the zeros of the Jacobi polynomial $P_n^{(α,β)}$ contains a sequence of constants $κ_1,κ_2,\\dots$, each of which is a polynomial in $α^2$ and $β^2$ with rational coefficients; for instance $κ_1=(α^2-β^2)/4$. This paper asks a simple arithmetic question: which primes, and to which powers, occur in the denominators of these coefficients? We prove that the odd part of the common denominator of $κ_r$ divides $\\operatorname{lcm}(1,3,\\ldots,2r-1)$, and that the power of $2$ is at most $2^{E_r}$ with $E_r=3r-1+ν_2((r-1)!)$, where $ν_2$ is the $2$-adic valuation. The study of the leading coefficient leads to the formula \\[ ν_2\\!\\left(\\sum_{j=0}^{m}\\binom mj\\frac1{2j+1}\\right)=m+ν_2(m+1)\\qquad(m\\ge0), \\] which we prove by showing that the series $\\sum_{k\\ge0}k!/(2k+1)!!$, whose real sum is $π/2$, converges to $0$ in the field $\\mathbb Q_2$ of $2$-adic numbers. Finally, we show that the statement ``the exponent $E_r$ is attained for every $r$'' is equivalent to an integrality property of the Taylor coefficients of a single explicit power series in one variable. That property is established in a separate paper, and is confirmed here independently, by exact computation, for $r\\le48$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.23006",
  "author_total": 1,
  "authors": [
    {
      "name": "Iván Area"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4yqxdpj8epvp3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.23006",
  "fulltext_url": null,
  "id": "4yqxdpj8epvp3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.23006",
  "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": "W7204179998",
  "pdf_url": "https://arxiv.org/pdf/2608.23006",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4yqxdpj8epvp3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2608.23006",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.23006"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Dyadic denominator bounds and a transfer principle for the phase constants of Jacobi zeros",
  "title_text": "Dyadic denominator bounds and a transfer principle for the phase constants of Jacobi zeros",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/4yqxdpj8epvp3",
  "venue": "arXiv",
  "versions": []
}