{
  "abstract": "We develop a methodology for designing infinite products of rational blocks whose exponents are polynomials in the index $k$. Matching power sums of the slot constants through order $n$ forces the Type-$N$ product with binomial exponent $\\binom{n+k-2}{n-1}$ to converge to a ratio of Vignéras multiple gamma values $Γ_n$; an analogue finder lifts any Type-1 evaluation to every higher type, and integer combinations of binomial exponents then realise arbitrary integer-valued polynomial exponents, yielding explicit products with exponents $k$, $k^2$, $k^3$, ... for constants such as $π/2$, $\\sqrt{2}$, $e^{2K/π}$, and rational multiples of $π^{M!}$ (Part I). As the main application (Part II) we prove that for every positive integer $n$, a finite multiple-gamma template $\\mathcal{S}_n$ with generalised Eulerian weights $T(n,k)$ (OEIS A225118) evaluates the Duke-Imamoğlu expression $\\mathcal{D}_n = β'(-n) + (\\log 4)\\,β(-n)$. For odd $n$ this yields a convergent Wallis-Eulerian product for $e^{β'(-n)}$; for even $n$ the raw product diverges. The proof expands the template through the multiple-gamma functional equation, evaluates the quarter-integer coefficients in closed form, and identifies the resulting Eulerian-binomial sums with Duke's polynomials $P_{n+1,\\ell}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29741",
  "author_total": 1,
  "authors": [
    {
      "name": "Joshua W. E. Farrell"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dt31gbw9ja7jt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29741",
  "fulltext_url": null,
  "id": "dt31gbw9ja7jt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29741",
  "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.CA"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7204957666",
  "pdf_url": "https://arxiv.org/pdf/2608.29741",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dt31gbw9ja7jt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29741",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29741"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Wallis-type products with polynomial exponents and the Dirichlet beta function at negative integers",
  "title_text": "Wallis-type products with polynomial exponents and the Dirichlet beta function at negative integers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/dt31gbw9ja7jt",
  "venue": "arXiv",
  "versions": []
}