{
  "abstract": "We associate a canonical operator to every linear recurrence sequence admitting a distinct-root Binet representation \\(\\mathrm{U}_n=\\sum_{j=1}^{k}A_jλ_j^n.\\) The construction is based on the spectral Binet operator \\(\\mathcal{B}_\\mathrm{U}=\\sum_{j=1}^{k}A_jT_{λ_j}, \\) where \\(T_{λ_j}f(x)=f(λ_jx)\\) is the dilation operator associated with the characteristic root \\(λ_j\\). The corresponding recurrence derivative is defined by \\(\\mathrm{D}_\\mathrm{U}=x^{-1}\\mathcal{B}_\\mathrm{U}.\\) Its fundamental property is the monomial identity \\(\\mathrm{D}_\\mathrm{U}(x^n)=\\mathrm{U}_nx^{n-1},\\) which establishes a direct correspondence between recurrence sequences and degree-lowering operators. We develop the operator theory of \\(\\mathrm{D}_\\mathrm{U}\\), including iteration, a spectral product rule, the associated Leibniz defect, and a quotient rule. The construction reveals a natural bifurcation determined by the value \\(\\mathrm{U}_0\\), producing the Ward and non-Ward regimes as intrinsic operatorial phenomena. The product structure is shown to be governed by \\(k-2\\) residual spectral channels, leading to notions of spectral cancellation and spectral balance. As an application, we obtain a complete qualitative classification of second-order recurrence calculi. Using Horadam sequences as a universal second-order model, we prove that every such calculus is determined by the invariant triple \\((\\mathrm{W}_0,Δ, Z_\\mathrm{W}),\\) where \\(\\mathrm{W}_0\\) identifies the Ward/non-Ward type, \\(Δ\\) determines the spectral geometry, and \\(Z_\\mathrm{W}\\) determines regularity or singularity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20593",
  "author_total": 1,
  "authors": [
    {
      "name": "Ronald Orozco López"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9h8vbmvs17h1w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20593",
  "fulltext_url": null,
  "id": "9h8vbmvs17h1w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20593",
  "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.FA"
    ],
    [
      "openalex",
      "3109|Advanced Mathematical Theories and Applications"
    ]
  ],
  "openalex": "W7222289120",
  "pdf_url": "https://arxiv.org/pdf/2610.20593",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9h8vbmvs17h1w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.20593",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20593"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The $\\mathrm{U}$-Derivative Associated with Linear Recurrence Sequences",
  "title_text": "The U-Derivative Associated with Linear Recurrence Sequences",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/9h8vbmvs17h1w",
  "venue": "arXiv",
  "versions": []
}