{
  "abstract": "Let $p_n^{(α,β)}$ denote the Jacobi polynomial orthonormal for the weight $(1-x)^α(1+x)^β$ on $[-1,1]$, where $α,β\\ge-1/2$, and put $S=α+β+1$. We prove the uniform degree--parameter estimate $$ (1-x)^{α+1/2}(1+x)^{β+1/2} \\bigl|p_n^{(α,β)}(x)\\bigr|^2 \\le C\\max\\bigl\\{1,S^{1/3},S^{1/2}(n+1)^{-1/6}\\bigr\\}. $$ This proves, in an equivalent symmetric parametrisation, the stronger degree-sensitive conjecture proposed by Krasikov and implies the Erdélyi--Magnus--Nevai conjecture. The proof starts from Krasikov's estimate in the high-parameter quadrant and transports it to the hard edges through weighted contiguous relations whose singular endpoint terms cancel; direct hypergeometric estimates control the remaining endpoint caps. A Bessel turning-point argument shows that the intermediate factor $S^{1/3}$ in the squared estimate cannot be omitted. We also derive degree-sensitive lower bounds for Jacobi Christoffel functions and Gauss--Jacobi quadrature weights.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30304",
  "author_total": 2,
  "authors": [
    {
      "name": "Qi-Feng Bai"
    },
    {
      "name": "Yu-Tian Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/az1ky75q9nms4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30304",
  "fulltext_url": null,
  "id": "az1ky75q9nms4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30304",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7204944986",
  "pdf_url": "https://arxiv.org/pdf/2608.30304",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/az1ky75q9nms4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30304",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30304"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The Erdélyi--Magnus--Nevai and Krasikov Conjectures for Jacobi Polynomials",
  "title_text": "The Erdélyi--Magnus--Nevai and Krasikov Conjectures for Jacobi Polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/az1ky75q9nms4",
  "venue": "arXiv",
  "versions": []
}