{
  "abstract": "We study orthogonal polynomials of $d = d_1+d_2$ variables with respect to a wrapped product weight function ${\\bm W}({\\bm x},{\\bm y}) = W_1({\\bm x}/ρ({\\bm y})) W_2(\\bm y)$ for $(\\bm x, \\bm y) \\in \\mathbb{R}^{d_1} \\times \\mathbb{R}^{d_2}$, where $ρ$ is either linear or the square root of a nonnegative quadratic polynomial, and identify all such polynomials that are eigenfunctions of a second-order linear differential operator. For $d =2$, it is known that there are primarily, up to affine transformations, five families of such polynomials, which are products or wrapped products of classical orthogonal polynomials of one variable; all five families have their counterparts in higher dimensions, but no characterization is known in dimension three or higher. Our study explores viable wrapped product families, finds explicit second-order differential operators for two new families of orthogonal polynomials in $d= d_1+d_2$ variables that have not been studied before if either $d_1>1$ or $d_2 > 1$, and provides, in particular, a complete list of such operators among all wrapped product orthogonal polynomials when $d = 3$. The list also includes four families that are eigenfunctions of a fourth-order differential operator, whereas no second-order operator is available. Moreover, orthogonal polynomials on the wrapped quadratic surfaces that are eigenvalues of a second-order differential operator on the surface are also studied.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01751",
  "author_total": 1,
  "authors": [
    {
      "name": "Yuan Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/em7f6xd62zjn4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01751",
  "fulltext_url": null,
  "id": "em7f6xd62zjn4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01751",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7207770202",
  "pdf_url": "https://arxiv.org/pdf/2609.01751",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/em7f6xd62zjn4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.01751",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01751"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Orthogonal polynomials which are eigenfunctions of a partial differential operator",
  "title_text": "Orthogonal polynomials which are eigenfunctions of a partial differential operator",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/em7f6xd62zjn4",
  "venue": "arXiv",
  "versions": []
}