{
  "abstract": "We investigate the integrality and structural properties of combinatorial sequences associated with the general Franel numbers $F_l^{(m)} = \\sum_{k=0}^l \\binom{l}{k}^m$ for exponents $m \\ge 3$. To avoid the emergence of fractional coefficients common in classical recurrence verification, we introduce an algebraic matrix framework operating entirely over the ring of integers $\\mathbb{Z}$. By constructing a unipotent transition matrix within a graded structure, we show that the intermediate variations satisfy specific boundary conditions under the matrix commutator. We prove that the underlying linear Diophantine system is strictly lower-triangular with unit diagonal entries, which ensures that the resulting coefficients are integers obtained via division-free back-substitution. This structural framework provides a matrix-theoretic formulation of the underlying divisibility properties, showing that the terminal component of the transformed vector reflects the global scalar invariant factorization $(T_{\\text{can}}P_{\\text{comb}}(l))_l = l^{3+m}\\Lambda_l$, where $\\Lambda_l \\in \\mathbb{Z}$. Explicit numerical examples for small dimensions are provided to illustrate the exactness of these integer relations over $\\mathbb{Z}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Abdellatif Aitelhad"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c6nedxnkt2ks8.bib",
  "cited_by": 0,
  "doi": "10.13140/rg.2.2.22039.53921/1",
  "fulltext_url": null,
  "id": "c6nedxnkt2ks8",
  "issns": [],
  "landing_url": "https://hal.science/hal-05635299v2",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ],
    [
      "openalex",
      "2604|Algebraic and Geometric Analysis"
    ]
  ],
  "openalex": "W7214113239",
  "pdf_url": "https://hal.science/hal-05635299/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c6nedxnkt2ks8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:01Z",
      "native_id": "hal-05635299",
      "source": "hal",
      "url": "https://hal.science/hal-05635299v2"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "A Matrix-Theoretic Representation and Integrality Properties of General Franel Sums",
  "title_text": "A Matrix-Theoretic Representation and Integrality Properties of General Franel Sums",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:01Z",
  "url": "https://research.draconitic.com/paper/c6nedxnkt2ks8",
  "venue": "HAL",
  "versions": []
}