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The aim of the present paper" /> <meta property="article:author" content="https://iste-tr.academia.edu/U%C4%9EURDURAN" /> <meta name="description" content="Recently, Araci et al. introduced (ρ, q)-analogue of the Haar distribution and by means of the distribution, they constructed (ρ, q)-Volkenborn integral yielding to Carlitz-type (ρ, q)-Bernoulli numbers and polynomials. The aim of the present paper" /> <title>(PDF) On (ρ, q)-Euler numbers and polynomials associated with (ρ, q)-Volkenborn integrals | UĞUR DURAN - Academia.edu</title> <link rel="canonical" href="https://www.academia.edu/84154884/On_%CF%81_q_Euler_numbers_and_polynomials_associated_with_%CF%81_q_Volkenborn_integrals" /> <script async src="https://www.googletagmanager.com/gtag/js?id=G-5VKX33P2DS"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-5VKX33P2DS', { cookie_domain: 'academia.edu', send_page_view: false, }); gtag('event', 'page_view', { 'controller': "single_work", 'action': "show", 'controller_action': 'single_work#show', 'logged_in': 'false', 'edge': 'unknown', // Send nil if there is no A/B test bucket, in case some records get logged // with missing data - that way we can distinguish between the two cases. // ab_test_bucket should be of the form <ab_test_name>:<bucket> 'ab_test_bucket': null, }) </script> <script> var $controller_name = 'single_work'; var $action_name = "show"; var $rails_env = 'production'; var $app_rev = '9387f500ddcbb8d05c67bef28a2fe0334f1aafb8'; var $domain = 'academia.edu'; var $app_host = "academia.edu"; var $asset_host = "academia-assets.com"; var $start_time = new Date().getTime(); var $recaptcha_key = "6LdxlRMTAAAAADnu_zyLhLg0YF9uACwz78shpjJB"; var $recaptcha_invisible_key = "6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj"; var $disableClientRecordHit = false; </script> <script> window.require = { config: function() { return function() {} } } </script> <script> window.Aedu = window.Aedu || {}; window.Aedu.hit_data = null; window.Aedu.serverRenderTime = new Date(1733020590000); window.Aedu.timeDifference = new Date().getTime() - 1733020590000; </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"ScholarlyArticle","abstract":"Recently, Araci et al. introduced (ρ, q)-analogue of the Haar distribution and by means of the distribution, they constructed (ρ, q)-Volkenborn integral yielding to Carlitz-type (ρ, q)-Bernoulli numbers and polynomials. The aim of the present paper is to introduce a generalization of the fermionic p-adic measure based on (ρ, q)-integers and set the corresponding integral to this measure. Consequently, Carlitz-type (ρ, q)-Euler polynomials and numbers are defined in terms of the above mentioned integral. 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distribution and by means of the distribution, they constructed (ρ, q)-Volkenborn integral yielding to Carlitz-type (ρ, q)-Bernoulli numbers and polynomials. The aim of the present paper is to introduce a generalization of the fermionic p-adic measure based on (ρ, q)-integers and set the corresponding integral to this measure. Consequently, Carlitz-type (ρ, q)-Euler polynomials and numbers are defined in terms of the above mentioned integral. 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The aim of the present paper is to introduce a generalization of the fermionic p-adic measure based on (ρ, q)-integers and set the corresponding integral to this measure. Consequently, Carlitz-type (ρ, q)-Euler polynomials and numbers are defined in terms of the above mentioned integral. Moreover, some of their identities and properties are established.</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":89275946,"attachmentType":"pdf","workUrl":"https://www.academia.edu/84154884/On_%CF%81_q_Euler_numbers_and_polynomials_associated_with_%CF%81_q_Volkenborn_integrals"}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":89275946,"attachmentType":"pdf","workUrl":"https://www.academia.edu/84154884/On_%CF%81_q_Euler_numbers_and_polynomials_associated_with_%CF%81_q_Volkenborn_integrals"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div></div><div data-auto_select="false" data-client_id="331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b" 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class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="990498" href="https://hku-tr.academia.edu/saraci">Serkan Araci</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of the Egyptian Mathematical Society (2014) xxx, xxx–xxx</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper, we focus on applications of q-Euler polynomials and obtain some new combinatorial relations by using p-adic q-integral on Zp. Moreover, we derive distribution formula (Multiplication Theorem) for Dirichlet type of q-Euler numbers and polynomials with weight a. Also we apply the method of analytic continuation of q-Euler polynomials which is the main result of this paper.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On the families of q-Euler numbers and polynomials and their applications","attachmentId":33758441,"attachmentType":"pdf","work_url":"https://www.academia.edu/1230871/On_the_families_of_q_Euler_numbers_and_polynomials_and_their_applications","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/1230871/On_the_families_of_q_Euler_numbers_and_polynomials_and_their_applications"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="7" data-entity-id="65429723" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/65429723/Explicit_formulas_for_p_adic_integrals_approach_to_p_adic_distributions_and_some_families_of_special_numbers_and_polynomials">Explicit formulas for p-adic integrals: approach to p-adic distributions and some families of special numbers and polynomials</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="146832572" href="https://independent.academia.edu/YilmazSimsek2">Yilmaz Simsek</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2019</p><p class="ds-related-work--abstract ds2-5-body-sm">The main objective of this article is to give and classify new formulas of $p$-adic integrals and blend these formulas with previously well known formulas. Therefore, this article gives briefly the formulas of $p$-adic integrals which were found previously, as well as applying the integral equations to the generating functions and other special functions, giving proofs of the new interesting and novel formulas. The $p$-adic integral formulas provided in this article contain several important well-known families of special numbers and special polynomials such as the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers, the Lah numbers, the Peters numbers and polynomials, the central factorial numbers, the Daehee numbers and polynomials, the Changhee numbers and polynomials, the Harmonic numbers, the Fubini numbers, combinatorial numbers and sums. In addition, we defined two new sequences containing the Bernoulli numbers and Euler numbers. These t...</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Explicit formulas for p-adic integrals: approach to p-adic distributions and some families of special numbers and polynomials","attachmentId":77030704,"attachmentType":"pdf","work_url":"https://www.academia.edu/65429723/Explicit_formulas_for_p_adic_integrals_approach_to_p_adic_distributions_and_some_families_of_special_numbers_and_polynomials","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/65429723/Explicit_formulas_for_p_adic_integrals_approach_to_p_adic_distributions_and_some_families_of_special_numbers_and_polynomials"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="8" data-entity-id="24681856" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/24681856/q_Genocchi_Numbers_and_Polynomials_Associated_with_Fermionic_p_Adic_Invariant_Integrals_on_Zp">q-Genocchi Numbers and Polynomials Associated with Fermionic p-Adic Invariant Integrals on ℤp</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="47577249" href="https://independent.academia.edu/LeechaeJang">Leechae Jang</a><span>, </span><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="39129151" href="https://independent.academia.edu/TaekyunKim3">Taekyun Kim</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Abstract and Applied Analysis, 2008</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"q-Genocchi Numbers and Polynomials Associated with Fermionic p-Adic Invariant Integrals on ℤp","attachmentId":45011421,"attachmentType":"pdf","work_url":"https://www.academia.edu/24681856/q_Genocchi_Numbers_and_Polynomials_Associated_with_Fermionic_p_Adic_Invariant_Integrals_on_Zp","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/24681856/q_Genocchi_Numbers_and_Polynomials_Associated_with_Fermionic_p_Adic_Invariant_Integrals_on_Zp"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="9" data-entity-id="61038332" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/61038332/On_p_adic_Gamma_Function_Related_to_q_Daehee_Polynomials_and_Numbers">On p-adic Gamma Function Related to q-Daehee Polynomials and Numbers</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="72285823" href="https://independent.academia.edu/A%C3%87IKG%C3%96ZM">Mehmet AÇIKGÖZ</a></div><p class="ds-related-work--abstract ds2-5-body-sm">In this paper, we investigate p-adic q-integral (q-Volkenborn integral) on ℤ_{p} of p-adic gamma function via their Mahler expansions. We also derived two q-Volkenborn integrals of p-adic gamma function in terms of q-Daehee polynomials and numbers and q-Daehee polynomials and numbers of the second kind. Moreover, we discover q-Volkenborn integral of the derivative of p-adic gamma function. We acquire the relationship between the p-adic gamma function and Stirling numbers of the first kind. We finally develop a novel and interesting representation for the p-adic Euler constant by means of the q-Daehee polynomials and numbers.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On p-adic Gamma Function Related to q-Daehee Polynomials and Numbers","attachmentId":74219511,"attachmentType":"pdf","work_url":"https://www.academia.edu/61038332/On_p_adic_Gamma_Function_Related_to_q_Daehee_Polynomials_and_Numbers","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/61038332/On_p_adic_Gamma_Function_Related_to_q_Daehee_Polynomials_and_Numbers"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div></div></div><div class="ds-sticky-ctas--wrapper js-loswp-sticky-ctas hidden"><div class="ds-sticky-ctas--grid-container"><div class="ds-sticky-ctas--container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--sticky-ctas","attachmentId":89275946,"attachmentType":"pdf","workUrl":null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--sticky-ctas","attachmentId":89275946,"attachmentType":"pdf","workUrl":null}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div><div class="ds-below-fold--grid-container"><div class="ds-work--container js-loswp-embedded-document"><div class="attachment_preview" data-attachment="Attachment_89275946" style="display: none"><div class="js-scribd-document-container"><div class="scribd--document-loading js-scribd-document-loader" style="display: block;"><img alt="Loading..." src="//a.academia-assets.com/images/loaders/paper-load.gif" /><p>Loading Preview</p></div></div><div style="text-align: center;"><div class="scribd--no-preview-alert js-preview-unavailable"><p>Sorry, preview is currently unavailable. 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js-related-work-sidebar-card" data-collection-position="11" data-entity-id="94429225" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/94429225/ON_THE_VOLKENBORN_INTEGRAL_OF_THE_q_EXTENSION_OF_THE_p_ADIC_GAMMA_FUNCTION">ON THE VOLKENBORN INTEGRAL OF THE q-EXTENSION OF THE p-ADIC GAMMA FUNCTION</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="112175718" href="https://mersin.academia.edu/HamzaMenken">Hamza Menken</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2017</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"ON THE VOLKENBORN INTEGRAL OF THE q-EXTENSION OF THE p-ADIC GAMMA 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data-author-id="146832572" href="https://independent.academia.edu/YilmazSimsek2">Yilmaz Simsek</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2021</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Interpolation Functions for New Classes Special Numbers and Polynomials via Applications of p-adic Integrals and Derivative Operator","attachmentId":77030718,"attachmentType":"pdf","work_url":"https://www.academia.edu/65429740/Interpolation_Functions_for_New_Classes_Special_Numbers_and_Polynomials_via_Applications_of_p_adic_Integrals_and_Derivative_Operator","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" 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data-author-id="241879659" href="https://independent.academia.edu/BayadAbdelmejid">Abdelmejid Bayad</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Inequalities and Applications, 2011</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A Study on the -Adic -Integral Representation on Associated with the Weighted -Bernstein and -Bernoulli Polynomials","attachmentId":104964778,"attachmentType":"pdf","work_url":"https://www.academia.edu/105537927/A_Study_on_the_Adic_Integral_Representation_on_Associated_with_the_Weighted_Bernstein_and_Bernoulli_Polynomials","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline 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