{
  "abstract": "We study signed generalized Stirling polynomials $P_k(m,x)$ arising in closed forms for Malmsten-type hyperbolic secant integrals. Their product structure is used to prove recurrences, gamma--polygamma formulas for $P_{m-s}(m,x)$, a central vanishing identity, a finite approximation to $\\cosh πx$, and a limit formula for $π$. We identify these polynomials as signed residues of the equal-period Barnes multiple zeta function, derive their reflection formula, and obtain finite parity-cancellation and Stirling cycle-number identities. We also compare the centered polynomials with a classical Meixner--Pollaczek orthogonal family. We evaluate finite nested sums built from the sequence $χ_n$. Fixing the lower bounds turns these sums into coefficient-counting problems: common lower bounds give binomial coefficients and staircase bounds give Catalan numbers. Combining these counts with known formulas for $χ_j$ yields explicit hyperbolic-secant integral evaluations involving Catalan's constant, zeta values, and polygamma values. A Wolfram Language package accompanies the formulas.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.26846",
  "author_total": 2,
  "authors": [
    {
      "name": "Abdulhafeez A. Abdulsalam"
    },
    {
      "name": "Michael J. Schlosser"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/913qetazj7wgx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.26846",
  "fulltext_url": null,
  "id": "913qetazj7wgx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.26846",
  "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.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2605.26846",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/913qetazj7wgx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2605.26846",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.26846"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Signed Generalized Stirling Polynomials, Nested Sums, and Hyperbolic Secant Integral Identities",
  "title_text": "Signed Generalized Stirling Polynomials, Nested Sums, and Hyperbolic Secant Integral Identities",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/913qetazj7wgx",
  "venue": "arXiv",
  "versions": []
}