{
  "abstract": "Let $π_n$ be the monic polynomial of degree $n$ orthogonal on $[-c,c]$, $0<c\\leq1$, with respect to the Jacobi weight $(1-x)^α(1+x)^β$, where $-1<α<β$. Gautschi conjectured that \\[ \\left[ \\frac{π_n(-c)}{π_n(c)} \\right]^2 \\left(\\frac{1-c}{1+c}\\right)^{β-α} <1. \\] By his variation formula, this inequality is sufficient for every positive zero of $π_n$ to move to the right as $c$ increases. For $0<c<1$, a first-crossing argument proves the conjecture throughout $0<α<β$. Together with the earlier regions recorded by Gautschi and established by Milovanović, this settles $β\\geq0$. In the negative wedge, writing $α=-r-λ$ and $β=-r+λ$, an ensemble Ward bound yields the region $c^2\\leq3/(3+r)$. A strengthening of the same crossing lemma, using an orthogonal expansion and Markov's theorem, removes this restriction. Consequently, the conjecture holds for every $n\\geq1$, $-1<α<β$, and $0<c\\leq1$. We also give a direct degree-one proof and an explicit asymptotic limit. The case $c=1$ is immediate.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.05963",
  "author_total": 3,
  "authors": [
    {
      "name": "V. Botta"
    },
    {
      "name": "K. Castillo"
    },
    {
      "name": "L. Tertuliano da Silva"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dbmj3cymyxsqg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.05963",
  "fulltext_url": null,
  "id": "dbmj3cymyxsqg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.05963",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2608.05963",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dbmj3cymyxsqg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2608.05963",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.05963"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A proof of Gautschi's conjecture on subrange Jacobi polynomials",
  "title_text": "A proof of Gautschi's conjecture on subrange Jacobi polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/dbmj3cymyxsqg",
  "venue": "arXiv",
  "versions": []
}