{
  "abstract": "Let $R_n^{(a,b)}=P_n^{(a,b)}/P_n^{(a,b)}(1)$ be the normalised Jacobi polynomials. For $α\\ge0$, put $ξ_0=-1$ and let $ξ_n$ be the largest zero of $R_n^{(α+1,α)}$ for $n\\ge1$. We prove that $$ R_k^{(α,α)}(x) =\\min_{n\\ge0}R_n^{(α,α)}(x), \\quad ξ_{k-1}\\le x\\leξ_k, $$ for every $k\\ge1$. This determines the pointwise lower envelope of the normalised ultraspherical polynomials and answers Question 2 posed by F. M. de Oliveira Filho in his 2009 thesis. For $α>0$ and $-1<x<1$, we also determine all minimising degrees. The sequence is non-increasing up to each such degree, which answers his Question 1 in this range. A Legendre example gives a negative answer when $α=0$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15473",
  "author_total": 2,
  "authors": [
    {
      "name": "K. Castillo"
    },
    {
      "name": "S. Sadigova"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ar3nj7v552ya0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15473",
  "fulltext_url": null,
  "id": "ar3nj7v552ya0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15473",
  "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": "W7213356360",
  "pdf_url": "https://arxiv.org/pdf/2609.15473",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ar3nj7v552ya0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.15473",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15473"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The lower envelope of ultraspherical polynomials",
  "title_text": "The lower envelope of ultraspherical polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/ar3nj7v552ya0",
  "venue": "arXiv",
  "versions": []
}