{
  "abstract": "In 2023, Yang introduced the Fibonacci grammar to study convolutions of Fibonacci and Lucas numbers. We develop this framework by incorporating the inverse operator D−1 and a sign-parametrized convolution grammar. These operators give grammatical expressions for Fibonacci-type numbers Tn(a,b) at arbitrary integral indices and lead to the Lucas dual Sn=Tn(2b−ap,bp+2a). We obtain unified addition and subtraction identities for this dual pair and a single representation covering ordinary and alternating binomial convolutions of both sequences. As an application, we correct the parity condition in a lemma of Kılıç, Ömür and Türker Ulutaş, which is valid only for odd scaling factors, and determine which of its consequences survive. We then construct a four-letter product grammar for the binomial convolutions of two generalized Fibonacci or Lucas numbers. Its two-channel form reorganizes the corresponding root evaluations and makes the parity dependence explicit. The same decomposition gives an arbitrary-weight formula; a scaling reduction treats positive second parameters, while a direct four-root form treats the real, distinct, nonzero-root Horadam regime. Finally, we extend the grammar to finitely many factors, prove the even-step parity criterion, and give the triple UUU formula as an explicit example. The genuinely additional structural results are the inverse-operator construction, the channel decomposition, the even-step criterion, and the triple-product formula; the overlapping closed evaluations are identified as equivalent reformulations of earlier root-based formulas.",
  "abstract_shown": true,
  "affiliations": [
    "Tianjin College, University of Science and Technology Beijing, Tianjin 301830, China",
    "School of Science, Tianjin University of Technology and Education, Tianjin 300222, China"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 3,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Dengxing Liang",
      "orcid": "0009-0001-2230-5193"
    },
    {
      "affs": [
        1
      ],
      "name": "Harold Ruilong Yang",
      "orcid": "0009-0004-1572-0673"
    },
    {
      "affs": [
        0
      ],
      "name": "Yanjun Zhang",
      "orcid": "0009-0000-4520-4435"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5ws55j2k2gdfq.bib",
  "cited_by": 0,
  "doi": "10.3390/math14203690",
  "fulltext_url": null,
  "id": "5ws55j2k2gdfq",
  "issns": [
    "2227-7390"
  ],
  "landing_url": "https://www.mdpi.com/2227-7390/14/20/3690",
  "language": null,
  "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": [
    [
      "openalex",
      "3109|Advanced Mathematical Theories and Applications"
    ]
  ],
  "openalex": "W7221510126",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": "MDPI AG",
  "ris": "https://research.draconitic.com/paper/5ws55j2k2gdfq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:29Z",
      "native_id": "10.3390/math14203690",
      "source": "crossref",
      "url": "https://www.mdpi.com/2227-7390/14/20/3690"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "openalex:3109|Advanced Mathematical Theories and Applications",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Grammatical Constructions for Binomial Convolutions of Fibonacci-Type Numbers and Their Products",
  "title_text": "Grammatical Constructions for Binomial Convolutions of Fibonacci-Type Numbers and Their Products",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:18:29Z",
  "url": "https://research.draconitic.com/paper/5ws55j2k2gdfq",
  "venue": "Mathematics",
  "versions": []
}