Degenerate Mittag-Leffler Functions Defined via the Degenerate Gamma Function and Applications to Fractional Maxwell-Zener Viscoelasticity

Degenerate Mittag-Leffler Functions Defined via the Degenerate Gamma Function and Applications to Fractional Maxwell-Zener Viscoelasticity | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Degenerate Mittag-Leffler Functions Defined via the Degenerate Gamma Function and Applications to Fractional Maxwell-Zener Viscoelasticity Oğuz Yağcı This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8844550/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Time-dependent materials often show relaxation and creep over many decades in time. Fractional Maxwell and Zener models describe this behavior with a small number of parameters, and their response functions are written in terms of Mittag--Leffler kernels. In this paper we introduce a $\lambda$--deformed two-parameter Mittag--Leffler function by replacing the classical gamma denominator in the Mittag--Leffler series with the degenerate gamma function $\Gamma_{\lambda}$. Using a Beta-integral representation of $\Gamma_{\lambda}$, we give admissible parameters and determine the exact radius of convergence $R_{\lambda}(\alpha)=|\lambda^{\alpha}|^{-1}$, which yields a sharp disk of analyticity. We also prove that $E^{(\lambda)}_{\alpha,\beta}$ converges to the classical Mittag--Leffler function acts as a memory-shape control that can improve fits to relaxation/creep data, while the standard fractional models are recovered in the limit $\lambda\to0^{+}$. 2020 Mathematics Subject Classification. 33E12; 33B15; 34A08; 74D05; 44A10; 26A33. degenerate gamma function degenerate Mittag–Leffler function fractional viscoelasticity fractional Maxwell model fractional Zener model Fox–Wright function Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8844550","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":589993393,"identity":"a1ade55d-7354-4161-b35b-3eb5e40f8d44","order_by":0,"name":"Oğuz Yağcı","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA80lEQVRIiWNgGAWjYDADNvbmgw+ANA8f8Vp4jiUbgLSwEW+NRI6aBFgvIYX8/GuMP/PU1MrzMeSwVX7NsZNhY2B++OgGHi2SM94YGPMcO27YxnD22G3ZbclAh7EZG+fg0WJw44wBUNmxBDbGvrTbktuYgVp42KQJaTnM8w+ohZnHrFhyWz0RWs73GDbzttUksLHxmDF+3HaYsBbJGWzFjHP7Dhi28bAlSzNuO87DxkzAL/z8hzd/ePOtTl5+/uODH39uq7bnZ29++BifFgaJBAYmHobDYDYzD5jEpxxszQEGxh8MdWA2kDEKRsEoGAWjABMAAO9zRHffkINSAAAAAElFTkSuQmCC","orcid":"","institution":"Kırıkkale University","correspondingAuthor":true,"prefix":"","firstName":"Oğuz","middleName":"","lastName":"Yağcı","suffix":""}],"badges":[],"createdAt":"2026-02-10 18:53:11","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8844550/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8844550/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":104399819,"identity":"ca8759d4-6454-4d33-92b4-5d4c92f27718","added_by":"auto","created_at":"2026-03-11 12:07:44","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":370628,"visible":true,"origin":"","legend":"","description":"","filename":"DML.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8844550/v1_covered_7e973700-b0df-47ee-9d45-c9a5d7720657.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eDegenerate Mittag-Leffler Functions Defined via the Degenerate Gamma Function and Applications to Fractional Maxwell-Zener Viscoelasticity\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"degenerate gamma function, degenerate Mittag–Leffler function, fractional viscoelasticity, fractional Maxwell model, fractional Zener model, Fox–Wright function","lastPublishedDoi":"10.21203/rs.3.rs-8844550/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8844550/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eTime-dependent materials often show relaxation and creep over many decades in time. Fractional Maxwell and Zener models describe this behavior with a small number of parameters, and their response functions are written in terms of Mittag--Leffler kernels. In this paper we introduce a $\\lambda$--deformed two-parameter Mittag--Leffler function by replacing the classical gamma denominator in the Mittag--Leffler series with the degenerate gamma function $\\Gamma_{\\lambda}$. Using a Beta-integral representation of $\\Gamma_{\\lambda}$, we give admissible parameters and determine the exact radius of convergence $R_{\\lambda}(\\alpha)=|\\lambda^{\\alpha}|^{-1}$, which yields a sharp disk of analyticity. 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