{
  "abstract": "Let $z_{n,j}(λ)$ denote the positive zeros, in decreasing order, of the ultraspherical polynomial $C_n^λ$, $λ>-1/2$, with the reduced limiting interpretation at $λ=0$ specified below. Our principal result settles three higher-monotonicity questions of Gautschi: two as printed and the natural open-interval form of the third, whose printed endpoint $λ=0$ is singular. For every $n\\geq3$, $$ \\sqrt{λ+1}\\,z_{n,j}(λ) $$ becomes a complete Bernstein function after translation of its parameter interval to $(0,\\infty)$. For every $n\\geq2$, $$ \\sqrtλ\\,z_{n,j}(λ) $$ is a complete Bernstein function on $(0,\\infty)$. For every $n\\geq4$, the largest-zero trajectory $$ \\sqrt{λ+\\frac{2n^2+1}{4n+2}}\\,z_{n,1}(λ) $$ has the same property, whilst for every other positive zero the derivative of this scaling fails to be completely monotone. Separately, an exact calculation in degree $4$ provides a counterexample to a fourth conjecture of Gautschi, concerning the linear scaling $λz_{n,j}(λ)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19186",
  "author_total": 1,
  "authors": [
    {
      "name": "K. Castillo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bqxdy166v8a7d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19186",
  "fulltext_url": null,
  "id": "bqxdy166v8a7d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19186",
  "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",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7213606055",
  "pdf_url": "https://arxiv.org/pdf/2609.19186",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bqxdy166v8a7d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.19186",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19186"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Complete Bernstein functions and scaled ultraspherical zeros",
  "title_text": "Complete Bernstein functions and scaled ultraspherical zeros",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/bqxdy166v8a7d",
  "venue": "arXiv",
  "versions": []
}