{
  "abstract": "Karp and Kuznetsov introduced the Meijer--Barnes $K$-function by replacing the Euler gamma factors in the Mellin--Barnes kernel of the Meijer $G$-function with Barnes double gamma factors, but excluded the regime $N=2(m+n)-p-q<0$. We complete this missing case. The kernel decays super-exponentially in two horizontal sectors, and the defining integral converges absolutely for every $z$ on the Riemann surface of the logarithm and every $α\\in\\mathbb{C}$. There are exactly four admissible contour classes, denoted $K^\\rightarrow,K^\\leftarrow,K^\\uparrow,K^\\downarrow$, satisfying the universal relation $K^\\uparrow+K^\\downarrow=K^\\rightarrow+K^\\leftarrow$ and hence spanning a space of dimension at most three. Under an explicit closing-arc growth hypothesis, we derive residue expansions and strengthened vanishing theorems; the three basic transformation laws hold unconditionally. We also prove that no vertical contour is admissible for $N<0$, replace the standard vertical-contour Mellin-transform method by a compatible two-index recurrence for residue coefficients, and develop a kernel-level insertion calculus together with analytic transform identities where the required interchange is justified, including integer and Riemann--Liouville fractional derivatives. Finally, we construct a numerical evaluator for the Barnes double gamma function at general $τ>0$ and validate representative identities and contour formulas to high precision.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16281",
  "author_total": 1,
  "authors": [
    {
      "name": "Changhao He"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/98m3tgf5ba64a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16281",
  "fulltext_url": null,
  "id": "98m3tgf5ba64a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16281",
  "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"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7213435002",
  "pdf_url": "https://arxiv.org/pdf/2609.16281",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/98m3tgf5ba64a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16281",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16281"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Completing the Meijer--Barnes $K$-function: the excluded case $N<0$",
  "title_text": "Completing the Meijer--Barnes K-function: the excluded case N<0",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/98m3tgf5ba64a",
  "venue": "arXiv",
  "versions": []
}