{
  "abstract": "We develop a unified calculus of fractional differential equations built on the k-deformed Gamma function &Gamma;_k, k > 0, in which the classical theory (k = 1), the ordinary exponential (α = k) and the half-order case (α = k/2) appear as special members of one family. We define a k-Riemann-Liouville integral, a k-Caputo derivative and a k-Laplace transform that diagonalizes them, and we show that the two-parameter k-Mittag-Leffler function E^k_{α,β} is the exact resolvent of the linear problem. This gives a variation-of-constants formula and closed forms in terms of exponentials, hyperbolic functions and e^(w&sup2;)erfc(−w). For nonlinear initial-value problems we prove global existence and uniqueness by the Banach principle in a Bielecki-type norm weighted by E^k_{α,k}, and existence under sublinear or small linear growth by Schauder's theorem. For a singular Dirichlet problem of order α &isin; (k, 2k] we show that the Green's function changes sign when α < 2k, so cone methods do not apply. We then derive k-analogues of the Euler reflection and Legendre duplication formulas and use them to evaluate the contraction thresholds in closed form. The reflection-case threshold is symmetric under γ ↦ 1−γ and maximal at γ = 1/2. All objects are exact rescalings of classical order-α/k objects, so the contribution is a unified parametrization with explicit k-dependent constants rather than a new function class. The identities and convergence claims are validated numerically.",
  "abstract_shown": true,
  "affiliations": [
    "College of Engineering Trivandrum"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Abhishek T V",
      "orcid": "0009-0006-2159-8584"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6ak2ag0pej30k.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23453972",
  "fulltext_url": null,
  "id": "6ak2ag0pej30k",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23453972",
  "language": "eng",
  "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": null,
  "pdf_url": "https://zenodo.org/api/records/23453972/files/k_Fractional_Mittag_Leffler_Paper_Final.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6ak2ag0pej30k.ris",
  "sources": [
    {
      "first_seen": "2026-11-01T12:21:04Z",
      "native_id": "23453972",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23453972"
    }
  ],
  "subjects": [],
  "title": "A k-Fractional Calculus with Mittag-Leffler Resolvents: Existence Theory and Closed-Form Contraction Thresholds for Singular Boundary-Value Problems",
  "title_text": "A k-Fractional Calculus with Mittag-Leffler Resolvents: Existence Theory and Closed-Form Contraction Thresholds for Singular Boundary-Value Problems",
  "type": "preprint",
  "updated_at": "2026-11-01T12:21:04Z",
  "url": "https://research.draconitic.com/paper/6ak2ag0pej30k",
  "venue": "Zenodo",
  "versions": []
}