{
  "abstract": "We study when a polynomial sequence $\\{p_n(x)\\}$ satisfying a linear homogeneous recurrence with polynomial coefficients admits an ordinary Riordan array as its coefficient matrix. For second-order recurrences $p_{n+2}(x)=a(x)p_{n+1}(x)+b(x)p_n(x)$, we give a complete characterization. We extend this to a necessary condition and a full factorization criterion for recurrences of arbitrary order $\\ell \\geq 3$. We then develop a production-matrix framework for arbitrary orthogonal polynomial sequences (OPS), not restricted to the Riordan-array-type case: for any lower-triangular invertible coefficient matrix $A$ of an OPS, the production matrix $P_{A^{-1}}=ASA^{-1}$ is always the tridiagonal Jacobi matrix encoding the three-term recurrence, and the first column of $A^{-1}$ always gives the moment sequence -- even when, as for the Legendre polynomials, the recurrence coefficients are non-constant and no Riordan array representation of the coefficient matrix exists. We illustrate this with Legendre and Chebyshev polynomials as, respectively, non-Riordan-type and Riordan-type examples, and give an explicit closed-form coefficient matrix and its inverse for the generalized Gegenbauer--Humbert OPS. Finally, we extend the framework to $d$-orthogonal polynomial sequences, showing that the coefficient matrix of a $d$-orthogonal sequence is always invertible, that its associated production matrix is $(d+2)$-banded (lower Hessenberg) and encodes the corresponding $(d+2)$-term recurrence, and that the first $d$ columns of the inverse matrix recover the moment sequences of the defining vector functional -- identifying $d$-orthogonality with $(d+1)$-Hessenberg generalized Riordan arrays and extending the classical $d=1$ tridiagonal correspondence.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28834",
  "author_total": 1,
  "authors": [
    {
      "name": "Tian-Xiao He"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/55jg54r034bhb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28834",
  "fulltext_url": null,
  "id": "55jg54r034bhb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28834",
  "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.CO"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7204938453",
  "pdf_url": "https://arxiv.org/pdf/2608.28834",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/55jg54r034bhb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.28834",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28834"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Riordan array representation of recursive polynomial sequences, orthogonal polynomial sequences, and $d$-orthogonal polynomial sequences",
  "title_text": "Riordan array representation of recursive polynomial sequences, orthogonal polynomial sequences, and d-orthogonal polynomial sequences",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/55jg54r034bhb",
  "venue": "arXiv",
  "versions": []
}