{
  "abstract": "For $p,q$ natural numbers, define $Δ^{\\langle p,q \\rangle}:=x^{p-1}\\frac{\\partial}{\\partial y} + y^{q-1}\\frac{\\partial}{\\partial x}$, and consider the sequence of polynomials $\\biggl((Δ^{\\langle p,q \\rangle})^n \\cdot x\\biggr)_{n\\geq 0}$; we call these the generalised hyperbolic polynomials. In this paper, we provide a simple combinatorial interpretation for these polynomials. As special cases, we obtain combinatorial interpretations for the coefficients of the analogs of sinh and cosh corresponding to the curve $x^p- p y^q/q=1$; on setting $p=q=3$, we obtain Conrad and Flajolet's tree interpretation for the coefficients of the Dixonian elliptic functions (2006) as a special case. Finally, we show that for any $p,q\\geq 3$, after suitably normalising followed by substituting $y=1$, we obtain a sequence of univariate polynomials which are real-rooted.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37931",
  "author_total": 1,
  "authors": [
    {
      "name": "Bishal Deb"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5fzczv9z1113e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37931",
  "fulltext_url": null,
  "id": "5fzczv9z1113e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37931",
  "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"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.37931",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5fzczv9z1113e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.37931",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37931"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Generalised hyperbolic polynomials: combinatorial interpretation and real-rootedness",
  "title_text": "Generalised hyperbolic polynomials: combinatorial interpretation and real-rootedness",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/5fzczv9z1113e",
  "venue": "arXiv",
  "versions": []
}