{
  "abstract": "We prove a binomial identity relating the $k$-Lucas and $k$-Fibonacci sequences: for real $k>0$ and integers $m\\ge1$, $n\\ge1$, a binomial-weighted sum of $k$-Lucas numbers reduces to a multiple of $L_{k,mn}$ when $n$ is even, but to a multiple of $F_{k,mn}$ when $n$ is odd. This parity dependence traces back to the $k$-Lucas Binet formula lacking the normalising factor $1/(α-β)$ present for $k$-Fibonacci numbers. The identity supplies, for each parity of $n$ and each $m$, an explicit family of positive weights summing to one; combined with the general annulus principle of Dalal and Govil, this yields a $k$-Lucas annulus containing all the zeros of a complex polynomial of degree $n$, complementing the $k$-Fibonacci-based bounds of Díaz-Barrero, Bidkham-Shashahani, and Kaur.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.26452",
  "author_total": 1,
  "authors": [
    {
      "name": "Herbert Batte"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5z09wq6ea2hva.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.26452",
  "fulltext_url": null,
  "id": "5z09wq6ea2hva",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.26452",
  "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.NT"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7204579956",
  "pdf_url": "https://arxiv.org/pdf/2608.26452",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5z09wq6ea2hva.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2608.26452",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.26452"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "$k$-Lucas Annulus for Polynomial Zeros",
  "title_text": "k-Lucas Annulus for Polynomial Zeros",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/5z09wq6ea2hva",
  "venue": "arXiv",
  "versions": []
}