{
  "abstract": "We study Verlinde sums $V_{n}(N,m)$, which for our purposes are finite sums of trigonometric functions $\\{r_{x}\\}$, associated to $\\textrm{ SU}(N)$. In this article we prove the following results: (i) the polynomial $D_N(w):=\\prod_x(w-r_x)$ is equal to, up to an explicit and computable factor, the iterated resultant of $Q(u)=u^m-1$ with an explicit elementary polynomial $W_N$ which is independent of $n$ and $m$; (ii) for fixed $N$ and $m$, $V_n(N,m)$ satisfies a linear recursion of order at most $\\binom{m-1}{N-1}$, together with arguments showing that one actually has a recursion of length equal to the number of distinct roots of $D_{N}$; (iii) for fixed $N$ and $m$, a Binet-type closed form expressing $V_n(N,m)$ as an explicit finite sum of $n$-th powers of algebraic numbers which are rescaled reciprocals of the distinct roots of $D_N(w)$ is derived. Several explicit examples are provided for $N=2,3,4,5,6$ and $8$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36088",
  "author_total": 3,
  "authors": [
    {
      "name": "Jay Jorgenson"
    },
    {
      "name": "Anders Karlsson"
    },
    {
      "name": "Lejla Smajlović"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/25jw2gr6pxxjh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36088",
  "fulltext_url": null,
  "id": "25jw2gr6pxxjh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36088",
  "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.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7214928283",
  "pdf_url": "https://arxiv.org/pdf/2609.36088",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/25jw2gr6pxxjh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.36088",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36088"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Resultants, Recursive Formulas and Binet Formulas for $\\textrm{SU}(N)$ Verlinde Sums",
  "title_text": "Resultants, Recursive Formulas and Binet Formulas for SU(N) Verlinde Sums",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/25jw2gr6pxxjh",
  "venue": "arXiv",
  "versions": []
}