Degenerate-Pochhammer-Based Truncated M-Fractional Operators with Applications to RC Relaxation

Degenerate-Pochhammer-Based Truncated M-Fractional Operators with Applications to RC Relaxation | 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-Pochhammer-Based Truncated M-Fractional Operators with Applications to RC Relaxation Oğuz Yağcı This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8576059/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 Real dielectric media and non-ideal circuit elements often exhibit relaxation behaviors that deviate from the classical exponential response of an ideal RC network.Fractional-order models have therefore become a standard tool for capturing anomalous (non-exponential) relaxation and for improving calibration against measured data. In this work, we introduce a degenerate-Pochhammer-based family of truncated $M$-fractional operators built from the degenerate gamma function and degenerate Pochhammer symbols.After a normalization that annihilates constants, we prove that the associated derivative is local on $C^{1}$ and admits the explicit representation\[{}^{i}\!D^{\alpha}_{M,\lambda} f(t)=K_{\lambda}\,t^{1-\alpha} f'(t),\qquad 0<\alpha\le 1,\]where $K_{\lambda}$ is an explicit constant determined by the first-order coefficient of the normalized kernel.In particular, for the parameter regime $p=q=0$, $\beta=\gamma=1$, and $i=1$, our construction recovers the conformable fractional derivative exactly. Using the operational time $s(t)=t^{\alpha}/(\alpha K_{\lambda})$, initial value problems written with ${}^{i}\!D^{\alpha}_{M,\lambda}$ reduce to ordinary differential equations in $s$.As a canonical application, we revisit the first-order resistor--capacitor (RC) relaxation model and obtain closed-form responses; for a step input $u(t)\equiv V_{0}$ we derive$v(t)=V_{0}+(v_{0}-V_{0})\exp\!\bigl(-t^{\alpha}/(\alpha K_{\lambda}RC)\bigr)$ and interpret $K_{\lambda}$ as an effective time-constant scaling.We also illustrate the reduction on lumped thermal dynamics and on damped/forced oscillators, and we include numerical examples. MSC Classification: 26A33 , 34A08 , 33B15 Fractional calculus truncated M-fractional operators degenerate gamma function degenerate Pochhammer symbols local fractional derivative fractional differential equations Full Text Additional Declarations No competing interests reported. Supplementary Files tablelambdaK.csv tableoscmetrics.csv tablercmetrics.csv tablethermalmetrics.csv 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. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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-8576059","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":573997221,"identity":"b796ac01-d565-412a-ae00-7c08a5b52c16","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":"2026-01-11 23:53:13","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8576059/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8576059/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":100646843,"identity":"9a7656d8-1007-480c-b796-b150ea33295b","added_by":"auto","created_at":"2026-01-20 05:01:41","extension":"eps","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":78184,"visible":true,"origin":"","legend":"","description":"","filename":"figoscforced.eps","url":"https://assets-eu.researchsquare.com/files/rs-8576059/v1/411fc8904f928474571e6443.eps"},{"id":100646969,"identity":"b4c1b4e1-27a1-4240-b2d7-5395ca1453cc","added_by":"auto","created_at":"2026-01-20 05:02:19","extension":"eps","order_by":1,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":49122,"visible":true,"origin":"","legend":"","description":"","filename":"figoscfree.eps","url":"https://assets-eu.researchsquare.com/files/rs-8576059/v1/db04e7d82fb00da1fb79a259.eps"},{"id":100646922,"identity":"83ca5896-43ff-4357-a4fd-f82a920eb8f3","added_by":"auto","created_at":"2026-01-20 05:02:00","extension":"pdf","order_by":2,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":564899,"visible":true,"origin":"","legend":"","description":"","filename":"snarticle.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8576059/v1/cbbea0b05b2f99fde81197c3.pdf"},{"id":100646841,"identity":"da0ef38b-56d7-411f-a040-6b7e1ec90786","added_by":"auto","created_at":"2026-01-20 05:01:39","extension":"eps","order_by":3,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":31649,"visible":true,"origin":"","legend":"","description":"","filename":"figrclambdaeffect.eps","url":"https://assets-eu.researchsquare.com/files/rs-8576059/v1/56f60a25827f7dbacb274c57.eps"},{"id":100646845,"identity":"5d8175d7-5cff-4af2-8c32-bf5f605c4378","added_by":"auto","created_at":"2026-01-20 05:01:41","extension":"eps","order_by":4,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":30734,"visible":true,"origin":"","legend":"","description":"","filename":"figrcstep.eps","url":"https://assets-eu.researchsquare.com/files/rs-8576059/v1/9c9801737bdd12fc24dc7a23.eps"},{"id":100646840,"identity":"80e5f958-43c6-4356-84fd-dbb7f75d75e1","added_by":"auto","created_at":"2026-01-20 05:01:39","extension":"eps","order_by":5,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":31385,"visible":true,"origin":"","legend":"","description":"","filename":"figthermalstep.eps","url":"https://assets-eu.researchsquare.com/files/rs-8576059/v1/93f74872de500bc830590136.eps"},{"id":100646943,"identity":"06bdb771-ec60-4a20-b44b-97f7dac43e28","added_by":"auto","created_at":"2026-01-20 05:02:12","extension":"csv","order_by":6,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":63,"visible":true,"origin":"","legend":"","description":"","filename":"tablelambdaK.csv","url":"https://assets-eu.researchsquare.com/files/rs-8576059/v1/a486afed94266d464d62527e.csv"},{"id":100646854,"identity":"022bd10b-a320-4201-bb30-ea1a120eceba","added_by":"auto","created_at":"2026-01-20 05:01:47","extension":"csv","order_by":7,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":261,"visible":true,"origin":"","legend":"","description":"","filename":"tableoscmetrics.csv","url":"https://assets-eu.researchsquare.com/files/rs-8576059/v1/9b3e5d236f10d34cef62c54f.csv"},{"id":100646862,"identity":"73342b2f-df81-40f9-942a-fbb28f86e28f","added_by":"auto","created_at":"2026-01-20 05:01:50","extension":"csv","order_by":8,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":325,"visible":true,"origin":"","legend":"","description":"","filename":"tablercmetrics.csv","url":"https://assets-eu.researchsquare.com/files/rs-8576059/v1/01838e3f2156a664901cd668.csv"},{"id":100646826,"identity":"9d622718-abda-43a2-b21d-ec02ecc233b2","added_by":"auto","created_at":"2026-01-20 05:01:25","extension":"csv","order_by":9,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":326,"visible":true,"origin":"","legend":"","description":"","filename":"tablethermalmetrics.csv","url":"https://assets-eu.researchsquare.com/files/rs-8576059/v1/84486d7bd24676bddd5a6451.csv"},{"id":100646859,"identity":"7d41a736-86c0-427b-af3d-eb6c5937bdd1","added_by":"auto","created_at":"2026-01-20 05:01:49","extension":"json","order_by":10,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":3511,"visible":true,"origin":"","legend":"","description":"","filename":"10c1aa82db364b22a364a1695c09f6a5.json","url":"https://assets-eu.researchsquare.com/files/rs-8576059/v1/1427deccaa0c9c1f51e4a86f.json"},{"id":107040360,"identity":"f7d04083-e633-4856-979c-6befcaf3149a","added_by":"auto","created_at":"2026-04-16 05:57:21","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":405238,"visible":true,"origin":"","legend":"","description":"","filename":"snarticle.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8576059/v1_covered_1e2249c7-a801-42bd-bedb-0e4086811eef.pdf"},{"id":100646967,"identity":"e1a36996-23f1-4066-b4f2-ef984a9d7a88","added_by":"auto","created_at":"2026-01-20 05:02:18","extension":"csv","order_by":0,"title":"","display":"","copyAsset":false,"role":"supplement","size":63,"visible":true,"origin":"","legend":"","description":"","filename":"tablelambdaK.csv","url":"https://assets-eu.researchsquare.com/files/rs-8576059/v1/0b42fa02da943d2502380323.csv"},{"id":100646927,"identity":"5d5be902-cda7-45d4-a29b-f82639b45b58","added_by":"auto","created_at":"2026-01-20 05:02:02","extension":"csv","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":261,"visible":true,"origin":"","legend":"","description":"","filename":"tableoscmetrics.csv","url":"https://assets-eu.researchsquare.com/files/rs-8576059/v1/ce1988ad7e795bed9c8b1cd9.csv"},{"id":100646909,"identity":"a6c9e786-93a9-47ce-bf1b-a400fbf56363","added_by":"auto","created_at":"2026-01-20 05:01:57","extension":"csv","order_by":2,"title":"","display":"","copyAsset":false,"role":"supplement","size":325,"visible":true,"origin":"","legend":"","description":"","filename":"tablercmetrics.csv","url":"https://assets-eu.researchsquare.com/files/rs-8576059/v1/3e82f5da2636fa8b0f15ed85.csv"},{"id":100646939,"identity":"35d8be7f-a9f6-45be-a35e-00e1a104e7ca","added_by":"auto","created_at":"2026-01-20 05:02:09","extension":"csv","order_by":3,"title":"","display":"","copyAsset":false,"role":"supplement","size":326,"visible":true,"origin":"","legend":"","description":"","filename":"tablethermalmetrics.csv","url":"https://assets-eu.researchsquare.com/files/rs-8576059/v1/5e01941fda696a132652a020.csv"}],"financialInterests":"No competing interests reported.","formattedTitle":"Degenerate-Pochhammer-Based Truncated M-Fractional Operators with Applications to RC Relaxation","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":"Fractional calculus, truncated M-fractional operators, degenerate gamma function, degenerate Pochhammer symbols, local fractional derivative, fractional differential equations","lastPublishedDoi":"10.21203/rs.3.rs-8576059/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8576059/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eReal dielectric media and non-ideal circuit elements often exhibit relaxation behaviors that deviate from the classical exponential response of an ideal RC network.Fractional-order models have therefore become a standard tool for capturing anomalous (non-exponential) relaxation and for improving calibration against measured data. In this work, we introduce a degenerate-Pochhammer-based family of truncated $M$-fractional operators built from the degenerate gamma function and degenerate Pochhammer symbols.After a normalization that annihilates constants, we prove that the associated derivative is local on $C^{1}$ and admits the explicit representation\\[{}^{i}\\!D^{\\alpha}_{M,\\lambda} f(t)=K_{\\lambda}\\,t^{1-\\alpha} f'(t),\\qquad 0\u0026lt;\\alpha\\le 1,\\]where $K_{\\lambda}$ is an explicit constant determined by the first-order coefficient of the normalized kernel.In particular, for the parameter regime $p=q=0$, $\\beta=\\gamma=1$, and $i=1$, our construction recovers the conformable fractional derivative exactly. Using the operational time $s(t)=t^{\\alpha}/(\\alpha K_{\\lambda})$, initial value problems written with ${}^{i}\\!D^{\\alpha}_{M,\\lambda}$ reduce to ordinary differential equations in $s$.As a canonical application, we revisit the first-order resistor--capacitor (RC) relaxation model and obtain closed-form responses; for a step input $u(t)\\equiv V_{0}$ we derive$v(t)=V_{0}+(v_{0}-V_{0})\\exp\\!\\bigl(-t^{\\alpha}/(\\alpha K_{\\lambda}RC)\\bigr)$ and interpret $K_{\\lambda}$ as an effective time-constant scaling.We also illustrate the reduction on lumped thermal dynamics and on damped/forced oscillators, and we include numerical examples.\u003c/p\u003e\n\u003cp\u003eMSC Classification: 26A33 , 34A08 , 33B15\u003c/p\u003e","manuscriptTitle":"Degenerate-Pochhammer-Based Truncated M-Fractional Operators with Applications to RC Relaxation","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-01-20 05:00:20","doi":"10.21203/rs.3.rs-8576059/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","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}}],"origin":"","ownerIdentity":"07e1c2ec-72cf-4e05-98f3-a0268a55822d","owner":[],"postedDate":"January 20th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-04-16T05:56:24+00:00","versionOfRecord":[],"versionCreatedAt":"2026-01-20 05:00:20","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8576059","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8576059","identity":"rs-8576059","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}