{
  "abstract": "We prove and substantially extend a conjecture of Gazeau, Josse-Michaux, and Monceau concerning the extreme positive zeros of Hermite polynomials, which arose from a finite-dimensional quantisation of the phase plane. For the generalised Laguerre polynomial $L_m^{(α)}$ of degree $m$ and parameter $α>-1$, the product of its $j$th smallest and $j$th largest zeros is strictly increasing with $m$ for each fixed positive integer $j$, once $m\\ge2j-1$. The classical Hermite-Laguerre identities then prove the original conjecture for the degree-$N$ Hermite polynomial $H_N$, in both parities and for every $N\\ge4$. As a further consequence, a one-parameter extension involving a deformation of the corresponding Jacobi matrix is settled.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08480",
  "author_total": 1,
  "authors": [
    {
      "name": "K. Castillo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/92hgd0dg4qqfj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08480",
  "fulltext_url": null,
  "id": "92hgd0dg4qqfj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08480",
  "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": "W7211993041",
  "pdf_url": "https://arxiv.org/pdf/2609.08480",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/92hgd0dg4qqfj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.08480",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08480"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Symmetric products of Laguerre zeros",
  "title_text": "Symmetric products of Laguerre zeros",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/92hgd0dg4qqfj",
  "venue": "arXiv",
  "versions": []
}