{
  "abstract": "This draft presents the construction of a two-variable extension E2sn(Θ,Ω)(ρ,y) of the Mittag–Leffler–Sheffer polynomials Esn(Θ,Ω)(ρ) by applying the heat (Gauss–Weierstrass) semigroup generated by the family’s own quasi-monomial lowering operator P^Es, rather than the ordinary derivative ∂ρ. This produces a one-parameter family of hybrid polynomials, an explicit finite-sum representation, and as the central result, a generalized evolution equation∂yE2sn=(P^Es)2E2sn in which the diffusion operator is determined by the Sheffer datum p and the Mittag–Leffler parameters (Θ,Ω); the datum q enters the initial data and the raising operator but not the evolution operator. We establish this identity as an exact identity on polynomials, and we show that it is a second-order equation in ρ precisely in the classical case p(t)=t, Θ=Ω=1: for p(t)=t and Θ∈N, the operator P^Es has a differential order of exactly Θ. The manuscript delivers a step-by-step proof of quasi-monomiality for the two-variable family via an explicit commutator computation (with an independent differential-operator sanity check), derives the corresponding raising operator and recursive formula, and verifies all results on three fully worked examples—two classical and one with Θ=2—supported by closed-form tables, three-dimensional surface plots, a finite-difference study of the classical reduction with a common final time and an observed-order column, a numerical test of the generalized operator itself, and MATLAB implementations.",
  "abstract_shown": true,
  "affiliations": [
    "Computer Science Department, College of Computer Sciences and Information Technology (CCSIT), King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia",
    "Department of Mathematics, Jamia Millia Islamia, New Delhi 110025, India"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 2,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Mohd. Sarfaraz",
      "orcid": "0000-0001-9968-9431"
    },
    {
      "affs": [
        1
      ],
      "name": "Umme Zainab",
      "orcid": "0000-0003-0582-9857"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1fyt7ys1dbr1r.bib",
  "cited_by": 0,
  "doi": "10.3390/math14203650",
  "fulltext_url": null,
  "id": "1fyt7ys1dbr1r",
  "issns": [
    "2227-7390"
  ],
  "landing_url": "https://www.mdpi.com/2227-7390/14/20/3650",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7221082869",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": "MDPI AG",
  "ris": "https://research.draconitic.com/paper/1fyt7ys1dbr1r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:29Z",
      "native_id": "10.3390/math14203650",
      "source": "crossref",
      "url": "https://www.mdpi.com/2227-7390/14/20/3650"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2604|Mathematical functions and polynomials",
      "subfield": "general"
    }
  ],
  "title": "Quasi-Monomiality and Heat-Semigroup Extensions of Mittag–Leffler–Sheffer Polynomials",
  "title_text": "Quasi-Monomiality and Heat-Semigroup Extensions of Mittag–Leffler–Sheffer Polynomials",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:18:29Z",
  "url": "https://research.draconitic.com/paper/1fyt7ys1dbr1r",
  "venue": "Mathematics",
  "versions": []
}