Derivatives of Srivastava’s triple hypergeometric functions

Derivatives of Srivastava’s triple hypergeometric 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 Derivatives of Srivastava’s triple hypergeometric functions Recep Şahin, Ayman Shehata, Shimaa I. Moustafa, Oğuz Yağcı This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8682619/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 We derive explicit closed-form expressions for the first-order derivatives with respect to all numerator and denominator parameters of Srivastava’s triple hypergeometric functionsHA,HB andHC. By differentiating the defining triple-series termwise and using standard identities for the Gamma and digamma functions together with Pochhammer-symbol relations, each parameter derivative is represented as a finite linear combination of Pathan’s quadruple hypergeometric function F(4) P with appropriately shifted parameter arrays. This unified F(4) P - based description provides a convenient framework for contiguous relations under unit shifts of the parameters and for sensitivity calculations. We also record Euler-type operator identities in the variables (x, y, z) and deduce the resulting systems of linear partial differential equations satisfied by HA, HB and HC. Numerical experiments based on truncated triple-series evaluation are included to corroborate the formulas and to illustrate parameter dependence; we specify the truncation bounds, finite-difference step sizes, and residual-type control criteria used in the computations. MSC Classification: 33C20 , 33C65 , 11J72 Srivastava triple hypergeometric functions parameter derivatives Pathan’s quadruple hypergeometric function contiguous relations Euler operators partial differential equations 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. 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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-8682619","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":580794016,"identity":"e745a761-3fa2-419c-8065-2ce5d8ae298a","order_by":0,"name":"Recep Şahin","email":"","orcid":"","institution":"Kırıkkale University","correspondingAuthor":false,"prefix":"","firstName":"Recep","middleName":"","lastName":"Şahin","suffix":""},{"id":580794017,"identity":"b98423d4-ed39-4bb7-94d9-1b407a3f033c","order_by":1,"name":"Ayman Shehata","email":"","orcid":"","institution":"Assiut University","correspondingAuthor":false,"prefix":"","firstName":"Ayman","middleName":"","lastName":"Shehata","suffix":""},{"id":580794018,"identity":"f8143105-ffa7-4266-9ebd-bb893f66af07","order_by":2,"name":"Shimaa I. 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By differentiating the defining triple-series termwise and using standard identities for the Gamma and digamma functions together with Pochhammer-symbol relations, each parameter derivative is represented as a finite linear combination of Pathan’s quadruple hypergeometric function F(4) P with appropriately shifted parameter arrays. This unified F(4) P - based description provides a convenient framework for contiguous relations under unit shifts of the parameters and for sensitivity calculations. We also record Euler-type operator identities in the variables (x, y, z) and deduce the resulting systems of linear partial differential equations satisfied by HA, HB and HC. Numerical experiments based on truncated triple-series evaluation are included to corroborate the formulas and to illustrate parameter dependence; we specify the truncation bounds, finite-difference step sizes, and residual-type control criteria used in the computations.\u003c/p\u003e\n\u003cp\u003eMSC Classification: 33C20 , 33C65 , 11J72\u003c/p\u003e","manuscriptTitle":"Derivatives of Srivastava’s triple hypergeometric functions","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-01-29 03:38:45","doi":"10.21203/rs.3.rs-8682619/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 29th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-01-29T03:38:45+00:00","versionOfRecord":[],"versionCreatedAt":"2026-01-29 03:38:45","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8682619","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8682619","identity":"rs-8682619","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}