Berndt-Type Integrals of Order Three and Series Associated with Jacobi Elliptic Functions

Berndt-Type Integrals of Order Three and Series Associated with Jacobi Elliptic Functions | 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 Berndt-Type Integrals of Order Three and Series Associated with Jacobi Elliptic Functions Hongyuan Rui, Ce Xu, Jianqiang Zhao This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3969711/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 In this paper, we first establish explicit evaluations of six classes of hyperbolic sums by special values of the Gamma function by using the tools of the Fourier series expansions and the Maclaurin series expansions of a few Jacobi elliptic functions developed in our previous paper. Then, using the method of contour integrations involving hyperbolic and trigonometric functions, we establish explicit evaluations of two families of Berndt-type integrals of order three by special values of the Gamma function. Furthermore, we present some interesting consequences and illustrative examples. AMS Subject Classifications (2020): 05A30, 32A27, 42A16, 33E05, 11B68. Berndt-type integral q-series hyperbolic and trigonometric functions contour integration Jacobi elliptic functions Fourier series expansions 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-3969711","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":274198386,"identity":"37ea1b2a-055c-4ee0-a40c-e25ced2ffc4e","order_by":0,"name":"Hongyuan Rui","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA00lEQVRIiWNgGAWjYDACCQbGBx8q/vEwtjc2PvxApBZmwxlnDsgw9xxuNpYgUgubMG/LARv2GeltAjzE6DCf3WPGwNtwh4d35sM2oH47Od0GAlpk7pwxeyC54xmP5OzEtgcFDMnGZgcIuUsix9zA8Awzj+HsxHYDCYYDiduI0GImkdjGzGN/82CbBA/RWg62HeZhnMFItJa0YsOGM2k8jD2JwEA2IMovyRsf/6mwsWdsP/7w4YcKOzmCWhgYOAyQOAY4lSED9gdEKRsFo2AUjIIRDABuL0Q7ORfDfAAAAABJRU5ErkJggg==","orcid":"","institution":"Anhui Normal University","correspondingAuthor":true,"prefix":"","firstName":"Hongyuan","middleName":"","lastName":"Rui","suffix":""},{"id":274198387,"identity":"ceaae531-dfec-4bbc-835a-8ebe10c97042","order_by":1,"name":"Ce Xu","email":"","orcid":"","institution":"Anhui Normal University","correspondingAuthor":false,"prefix":"","firstName":"Ce","middleName":"","lastName":"Xu","suffix":""},{"id":274198388,"identity":"dd2e1d1e-2576-46bb-a457-7bb97e2ff7bf","order_by":2,"name":"Jianqiang Zhao","email":"","orcid":"","institution":"The Bishop's School","correspondingAuthor":false,"prefix":"","firstName":"Jianqiang","middleName":"","lastName":"Zhao","suffix":""}],"badges":[],"createdAt":"2024-02-19 10:20:44","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3969711/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3969711/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":54633431,"identity":"806ec31c-34ac-48be-adf0-8590ff9734ec","added_by":"auto","created_at":"2024-04-13 21:22:37","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":264321,"visible":true,"origin":"","legend":"","description":"","filename":"RXZ010120242.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3969711/v1_covered_ecfa04a8-5fbc-42c8-a76c-ac73c4753a02.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Berndt-Type Integrals of Order Three and Series Associated with Jacobi Elliptic Functions","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"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":"Berndt-type integral, q-series, hyperbolic and trigonometric functions, contour integration, Jacobi elliptic functions, Fourier series expansions","lastPublishedDoi":"10.21203/rs.3.rs-3969711/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3969711/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn this paper, we first establish explicit evaluations of six classes of hyperbolic sums by special values of the Gamma function by using the tools of the Fourier series expansions and the Maclaurin series expansions of a few Jacobi elliptic functions developed in our previous paper. 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