{
  "abstract": "For the Lucas numbers $L_0=2$, $L_1=1$, and $L_n=L_{n-1}+L_{n-2}$, we obtain \\[ \\sum_{\\substack{n,m\\ge1\\\\(n,m)=1}}\\frac{1}{L_{2n}L_{2m}L_{2(n+m)}}=\\frac1{45}. \\] This follows from the absolutely convergent identity \\[ \\sum_{\\substack{n,m\\ge1\\\\(n,m)=1}}\\frac{q^{n+m}} {(1+q^n)(1+q^m)(1+q^{n+m})} =\\frac{q^2}{(1-q^2)^2},\\qquad |q|<1. \\] We give a self-contained proof by local cancellation on the Calkin--Wilf tree, as a specialization of Hata's Farey-interval telescoping principle. Specializations yield a hyperbolic-cosine identity \\[\\sum_{\\substack{n,m\\ge1\\\\(n,m)=1}} \\frac{1}{\\cosh(nx)\\cosh(mx)\\cosh((n+m)x)} =\\frac{2}{\\sinh^2(2x)}, x>0\\] and weighted sums for canonical second-order Lucas recurrences. Decomposing arbitrary positive pairs according to their greatest common divisor gives an unrestricted sum expressed by a Lambert series and the quasimodular Eisenstein series $E_2$. Conversely, Möbius inversion recovers the primitive identity from a double Lambert-series result of Cui, Kumar, and Singh.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26949",
  "author_total": 1,
  "authors": [
    {
      "name": "Nikita Kalinin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/czprmbs6tn4z9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26949",
  "fulltext_url": null,
  "id": "czprmbs6tn4z9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26949",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GM"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7223092185",
  "pdf_url": "https://arxiv.org/pdf/2610.26949",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/czprmbs6tn4z9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.26949",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26949"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Advanced Mathematical Identities",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Lucas recurrences and Lambert series in coprime double sums",
  "title_text": "Lucas recurrences and Lambert series in coprime double sums",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/czprmbs6tn4z9",
  "venue": "arXiv",
  "versions": []
}