{
  "abstract": "Let \\(g_n(x)\\) be the Mittag--Leffler polynomials defined by $$ \\sum_{n\\ge0}g_n(x)t^n=\\frac12\\left(\\frac{1+t}{1-t}\\right)^x. $$ We derive the coefficient formula $$ g_n(x)=\\frac1n\\sum_{j=0}^{\\lfloor (n-1)/2\\rfloor} (-1)^j\\frac{2^{n-2j-1}}{(n-2j-1)!} [u^{2j}](u\\cot u)^n\\,x^{n-2j}, $$ equivalently $$ [u^{n-r}](u\\cot u)^n =(-1)^{(n-r)/2}\\frac{n(r-1)!}{2^{r-1}}[x^r]g_n(x). $$ The identity is suggested by comparing an umbral representation of the hyperbolic tangent integral $$ K(a,b)=\\int_0^\\infty \\frac{\\tanh^a x}{x^b}\\,dx $$ with a compact cotangent-coefficient formula, but is proved independently from the generating function by formal Lagrange--Bürmann inversion. Substitution of the proved bridge back into the integral formula then yields a rigorous derivation of the umbral Mittag--Leffler representation. We also develop a Dirichlet--beta analogue for $$ I_{m,n}^β=\\int_0^\\infty\\frac{\\tanh^m x}{x^n\\cosh x}\\,dx. $$ Defining \\(q_m(x)\\) by $$ \\sum_{m\\ge0}q_m(x)t^m=(1-t^2)^{-1/2}\\exp(2x\\operatorname{artanh}t), $$ we prove $$ [u^{m-r}]\\frac{u}{\\sin u}(u\\cot u)^m =(-1)^{(m-r)/2}\\frac{r!}{2^r}[x^r]q_m(x). $$ This gives a parallel umbral representation for the beta integrals. The family \\(q_m\\) is a shifted \\(c=-1,β=1\\) specialization of the classical Meixner family, yielding a beta/Meixner counterpart to the zeta/Mittag--Leffler correspondence.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16092",
  "author_total": 1,
  "authors": [
    {
      "name": "Luc Ramsès Talla Waffo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/015zk5865x7ec.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16092",
  "fulltext_url": null,
  "id": "015zk5865x7ec",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16092",
  "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.GM"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7213428442",
  "pdf_url": "https://arxiv.org/pdf/2609.16092",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/015zk5865x7ec.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16092",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16092"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Advanced Mathematical Identities",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "From Umbral Hyperbolic Integrals to a Cotangent Coefficient Formula for the Mittag Leffler Polynomials",
  "title_text": "From Umbral Hyperbolic Integrals to a Cotangent Coefficient Formula for the Mittag Leffler Polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/015zk5865x7ec",
  "venue": "arXiv",
  "versions": []
}