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By using these generating functions, we introduce not only fundamental properties of these combinatorial numbers and polynomials, but also new identities and formulas. In general, identities and formulas obtained in this paper include the newly introduced combinatorial numbers and polynomials, Bernoulli numbers and polynomials, Euler numbers and polynomials, Apostol-Bernoulli numbers and polynomials, Apostol-Euler numbers and polynomials, Stirling numbers of the second kind, Daehee numbers, Changhee numbers, the generalized Eulerian type numbers, Eulerian polynomials, Fubini numbers, Dobinski numbers. Moreover, by applying derivative operator to the generating functions for tw...","author":[{"@context":"https://schema.org","@type":"Person","name":"Yilmaz Simsek"}],"contributor":[],"dateCreated":"2021-12-21","dateModified":null,"datePublished":"2021-01-01","headline":"Interpolation Functions for New Classes Special Numbers and Polynomials via Applications of p-adic Integrals and Derivative Operator","inLanguage":"en","keywords":[],"locationCreated":null,"publication":null,"publisher":{"@context":"https://schema.org","@type":"Organization","name":null},"image":null,"thumbnailUrl":null,"url":"https://www.academia.edu/65429740/Interpolation_Functions_for_New_Classes_Special_Numbers_and_Polynomials_via_Applications_of_p_adic_Integrals_and_Derivative_Operator","sourceOrganization":[{"@context":"https://schema.org","@type":"EducationalOrganization","name":null}]}</script><link rel="stylesheet" media="all" 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{"work":{"id":65429740,"created_at":"2021-12-21T23:44:09.508-08:00","from_world_paper_id":188469117,"updated_at":"2021-12-22T00:07:02.460-08:00","_data":{"abstract":"The main purpose of this paper is to not only define Apostol type new classes of numbers and polynomials, but also construct generating function for two new classes of special combinatorial numbers and polynomials by applications of p-adic integrals including the Volkenborn integral and the fermionic integral. By using these generating functions, we introduce not only fundamental properties of these combinatorial numbers and polynomials, but also new identities and formulas. In general, identities and formulas obtained in this paper include the newly introduced combinatorial numbers and polynomials, Bernoulli numbers and polynomials, Euler numbers and polynomials, Apostol-Bernoulli numbers and polynomials, Apostol-Euler numbers and polynomials, Stirling numbers of the second kind, Daehee numbers, Changhee numbers, the generalized Eulerian type numbers, Eulerian polynomials, Fubini numbers, Dobinski numbers. Moreover, by applying derivative operator to the generating functions for tw...","publication_date":"2021,,"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Interpolation Functions for New Classes Special Numbers and Polynomials via Applications of p-adic Integrals and Derivative Operator","broadcastable":false,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [146832572]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "control"; window.loswp.useOptimizedScribd4genScript = false; window.loswp.appleClientId = 'edu.academia.applesignon';</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":77030718,"attachmentType":"pdf"}"><img alt="First page of “Interpolation Functions for New Classes Special Numbers and Polynomials via Applications of p-adic Integrals and Derivative Operator”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/77030718/mini_magick20211221-17385-1grz5v6.png?1640159148" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/assets/single_work_splash/adobe.icon-574afd46eb6b03a77a153a647fb47e30546f9215c0ee6a25df597a779717f9ef.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">Interpolation Functions for New Classes Special Numbers and Polynomials via Applications of p-adic Integrals and Derivative Operator</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="146832572" href="https://independent.academia.edu/YilmazSimsek2"><img alt="Profile image of Yilmaz Simsek" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />Yilmaz Simsek</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2021</p><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">24 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 65429740; const worksViewsPath = "/v0/works/views?subdomain_param=api&work_ids%5B%5D=65429740"; const getWorkViews = async (workId) => { const response = await fetch(worksViewsPath); if (!response.ok) { throw new Error('Failed to load work views'); } const data = await response.json(); return data.views[workId]; }; // Get the view count for the work - we send this immediately rather than waiting for // the DOM to load, so it can be available as soon as possible (but without holding up // the backend or other resource requests, because it's a bit expensive and not critical). const viewCount = await getWorkViews(workId); const updateViewCount = (viewCount) => { const viewCountNumber = Number(viewCount); if (!viewCountNumber) { throw new Error('Failed to parse view count'); } const commaizedViewCount = viewCountNumber.toLocaleString(); const viewCountBody = document.getElementById('work-metadata-view-count'); if (viewCountBody) { viewCountBody.textContent = `${commaizedViewCount} views`; } else { throw new Error('Failed to find work views element'); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">The main purpose of this paper is to not only define Apostol type new classes of numbers and polynomials, but also construct generating function for two new classes of special combinatorial numbers and polynomials by applications of p-adic integrals including the Volkenborn integral and the fermionic integral. By using these generating functions, we introduce not only fundamental properties of these combinatorial numbers and polynomials, but also new identities and formulas. In general, identities and formulas obtained in this paper include the newly introduced combinatorial numbers and polynomials, Bernoulli numbers and polynomials, Euler numbers and polynomials, Apostol-Bernoulli numbers and polynomials, Apostol-Euler numbers and polynomials, Stirling numbers of the second kind, Daehee numbers, Changhee numbers, the generalized Eulerian type numbers, Eulerian polynomials, Fubini numbers, Dobinski numbers. Moreover, by applying derivative operator to the generating functions for tw...</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":77030718,"attachmentType":"pdf","workUrl":"https://www.academia.edu/65429740/Interpolation_Functions_for_New_Classes_Special_Numbers_and_Polynomials_via_Applications_of_p_adic_Integrals_and_Derivative_Operator"}">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":77030718,"attachmentType":"pdf","workUrl":"https://www.academia.edu/65429740/Interpolation_Functions_for_New_Classes_Special_Numbers_and_Polynomials_via_Applications_of_p_adic_Integrals_and_Derivative_Operator"}"><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" data-doc_id="77030718" data-landing_url="https://www.academia.edu/65429740/Interpolation_Functions_for_New_Classes_Special_Numbers_and_Polynomials_via_Applications_of_p_adic_Integrals_and_Derivative_Operator" data-login_uri="https://www.academia.edu/registrations/google_one_tap" data-moment_callback="onGoogleOneTapEvent" id="g_id_onload"></div><div class="ds-top-related-works--grid-container"><div class="ds-related-content--container ds-top-related-works--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="0" data-entity-id="65429680" 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/65429680/Analysis_of_the_p_adic_q_Volkenborn_integrals_An_approach_to_generalized_Apostol_type_special_numbers_and_polynomials_and_their_applications">Analysis of the p-adic q-Volkenborn integrals: An approach to generalized Apostol-type special numbers and polynomials and their applications</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">Cogent Mathematics</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":"Analysis of the p-adic q-Volkenborn integrals: An approach to generalized Apostol-type special numbers and polynomials and their applications","attachmentId":77030805,"attachmentType":"pdf","work_url":"https://www.academia.edu/65429680/Analysis_of_the_p_adic_q_Volkenborn_integrals_An_approach_to_generalized_Apostol_type_special_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/65429680/Analysis_of_the_p_adic_q_Volkenborn_integrals_An_approach_to_generalized_Apostol_type_special_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="1" data-entity-id="77100416" 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/77100416/New_Families_of_Special_Polynomial_Identities_Based_upon_Combinatorial_Sums_Related_to_p_Adic_Integrals">New Families of Special Polynomial Identities Based upon Combinatorial Sums Related to p-Adic Integrals</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">Symmetry, 2021</p><p class="ds-related-work--abstract ds2-5-body-sm">The aim of this paper is to study and investigate generating-type functions, which have been recently constructed by the author, with the aid of the Euler’s identity, combinatorial sums, and p-adic integrals. Using these generating functions with their functional equation, we derive various interesting combinatorial sums and identities including new families of numbers and polynomials, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Daehee numbers, the Changhee numbers, and other numbers and polynomials. 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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="5" data-entity-id="77100392" 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/77100392/Identities_and_relations_for_Fubini_type_numbers_and_polynomials_via_generating_functions_and_p_adic_integral_approach">Identities and relations for Fubini type numbers and polynomials via generating functions and p-adic integral approach</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">Publications de l'Institut Math?matique (Belgrade), 2019</p><p class="ds-related-work--abstract ds2-5-body-sm">The Fubini type polynomials have many application not only especially in combinatorial analysis, but also other branches of mathematics, in engineering and related areas. Therefore, by using the p-adic integrals method and functional equation of the generating functions for Fubini type polynomials and numbers, we derive various different new identities, relations and formulas including well-known numbers and polynomials such as the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers of the second kind, the ?-array polynomials and the Lah 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":"Identities and relations for Fubini type numbers and polynomials via generating functions and p-adic integral approach","attachmentId":84580090,"attachmentType":"pdf","work_url":"https://www.academia.edu/77100392/Identities_and_relations_for_Fubini_type_numbers_and_polynomials_via_generating_functions_and_p_adic_integral_approach","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/77100392/Identities_and_relations_for_Fubini_type_numbers_and_polynomials_via_generating_functions_and_p_adic_integral_approach"><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="6" data-entity-id="51060558" 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/51060558/Identities_related_to_special_polynomials_and_combinatorial_numbers">Identities related to special polynomials and combinatorial 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="146832572" href="https://independent.academia.edu/YilmazSimsek2">Yilmaz Simsek</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Filomat</p><p class="ds-related-work--abstract ds2-5-body-sm">The aim of this paper is to give some new identities and relations related to the some families of special numbers such as the Bernoulli numbers, the Euler numbers, the Stirling numbers of the first and second kinds, the central factorial numbers and also the numbers y1(n,k,?) and y2(n,k,?) which are given Simsek [31]. Our method is related to the functional equations of the generating functions and the fermionic and bosonic p-adic Volkenborn integral on Zp. Finally, we give remarks and comments on our results.</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":"Identities related to special polynomials and combinatorial numbers","attachmentId":68920589,"attachmentType":"pdf","work_url":"https://www.academia.edu/51060558/Identities_related_to_special_polynomials_and_combinatorial_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/51060558/Identities_related_to_special_polynomials_and_combinatorial_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 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="65429719" 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/65429719/Derivation_of_Computational_Formulas_for_certain_class_of_finite_sums_Approach_to_Generating_functions_arising_from_p_adic_integrals_and_special_functions">Derivation of Computational Formulas for certain class of finite sums: Approach to Generating functions arising from $p$-adic integrals and special functions</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">2021</p><p class="ds-related-work--abstract ds2-5-body-sm">The aim of this paper is to construct generating functions for some families of special finite sums with the aid of the Newton-Mercator series, hypergeometric series, and p-adic integral (the Volkenborn integral). By using these generating functions, their functional equations, and their partial derivative equations, many novel computational formulas involving the special finite sums of (inverse) binomial coefficients, the Bernoulli type polynomials and numbers, Euler polynomials and numbers, the Stirling numbers, the (alternating) harmonic numbers, the Leibnitz polynomials and others. Among these formulas, by considering a computational formula which computes the aforementioned certain class of finite sums with the aid of the Bernoulli numbers and the Stirling numbers of the first kind, we present a computation algorithm and we provide some of their special values. Morover, using the aforementioned special finite sums and combinatorial numbers, we give relations among multiple alte...</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":"Derivation of Computational Formulas for certain class of finite sums: Approach to Generating functions arising from $p$-adic integrals and special functions","attachmentId":77030710,"attachmentType":"pdf","work_url":"https://www.academia.edu/65429719/Derivation_of_Computational_Formulas_for_certain_class_of_finite_sums_Approach_to_Generating_functions_arising_from_p_adic_integrals_and_special_functions","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/65429719/Derivation_of_Computational_Formulas_for_certain_class_of_finite_sums_Approach_to_Generating_functions_arising_from_p_adic_integrals_and_special_functions"><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="65429718" 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/65429718/Formulas_for_p_adic_q_integrals_including_falling_rising_factorials_combinatorial_sums_and_special_numbers">Formulas for p-adic q-integrals including falling-rising factorials, combinatorial sums and special 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="146832572" href="https://independent.academia.edu/YilmazSimsek2">Yilmaz Simsek</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2017</p><p class="ds-related-work--abstract ds2-5-body-sm">The main purpose of this paper is to provide a novel approach to deriving formulas for the p-adic q-integral including the Volkenborn integral and the p-adic fermionic integral. By applying integral equations and these integral formulas to the falling factorials, the rising factorials and binomial coefficients, we derive some new and old identities and relations related to various combinatorial sums, well-known special numbers such as the Bernoulli and Euler numbers, the harmonic numbers, the Stirling numbers, the Lah numbers, the Harmonic numbers, the Fubini numbers, the Daehee numbers and the Changhee numbers. Applying these identities and formulas, we give some new combinatorial sums. Finally, by using integral equations, we derive generating functions for new families of special numbers and polynomials. We also give further comments and remarks on these functions, numbers and integral formulas.</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":"Formulas for p-adic q-integrals including falling-rising factorials, combinatorial sums and special numbers","attachmentId":77030708,"attachmentType":"pdf","work_url":"https://www.academia.edu/65429718/Formulas_for_p_adic_q_integrals_including_falling_rising_factorials_combinatorial_sums_and_special_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/65429718/Formulas_for_p_adic_q_integrals_including_falling_rising_factorials_combinatorial_sums_and_special_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":77030718,"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":77030718,"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_77030718" 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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data-author-id="146832572" href="https://independent.academia.edu/YilmazSimsek2">Yilmaz Simsek</a></div><p class="ds-related-work--metadata ds2-5-body-xs">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":"Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications","attachmentId":84580104,"attachmentType":"pdf","work_url":"https://www.academia.edu/77100428/Generating_functions_for_generalized_Stirling_type_numbers_Array_type_polynomials_Eulerian_type_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-related-work-grid-card-view-pdf" href="https://www.academia.edu/77100428/Generating_functions_for_generalized_Stirling_type_numbers_Array_type_polynomials_Eulerian_type_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-related-work-sidebar-card" data-collection-position="5" data-entity-id="65429695" 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/65429695/Identities_and_derivative_formulas_for_the_combinatorial_and_Apostol_Euler_type_numbers_by_their_generating_functions">Identities and derivative formulas for the combinatorial and Apostol-Euler type numbers by their generating functions</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="146832572" href="https://independent.academia.edu/YilmazSimsek2">Yilmaz Simsek</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Filomat</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":"Identities and derivative formulas for the combinatorial and Apostol-Euler type numbers by their generating functions","attachmentId":77030812,"attachmentType":"pdf","work_url":"https://www.academia.edu/65429695/Identities_and_derivative_formulas_for_the_combinatorial_and_Apostol_Euler_type_numbers_by_their_generating_functions","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" href="https://www.academia.edu/65429695/Identities_and_derivative_formulas_for_the_combinatorial_and_Apostol_Euler_type_numbers_by_their_generating_functions"><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-related-work-sidebar-card" data-collection-position="6" data-entity-id="51060567" 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/51060567/Some_New_Families_of_Special_Polynomials_and_Numbers_Associated_with_Finite_Operators">Some New Families of Special Polynomials and Numbers Associated with Finite Operators</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="146832572" href="https://independent.academia.edu/YilmazSimsek2">Yilmaz Simsek</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Symmetry</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":"Some New Families of Special Polynomials and Numbers Associated with Finite Operators","attachmentId":68920528,"attachmentType":"pdf","work_url":"https://www.academia.edu/51060567/Some_New_Families_of_Special_Polynomials_and_Numbers_Associated_with_Finite_Operators","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" href="https://www.academia.edu/51060567/Some_New_Families_of_Special_Polynomials_and_Numbers_Associated_with_Finite_Operators"><span class="ds2-5-text-link__content">View 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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":"Construction of some new families of Apostol-type numbers and polynomials via Dirichlet character and p-adic q-integrals","attachmentId":77030802,"attachmentType":"pdf","work_url":"https://www.academia.edu/65429688/Construction_of_some_new_families_of_Apostol_type_numbers_and_polynomials_via_Dirichlet_character_and_p_adic_q_integrals","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" href="https://www.academia.edu/65429688/Construction_of_some_new_families_of_Apostol_type_numbers_and_polynomials_via_Dirichlet_character_and_p_adic_q_integrals"><span 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data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A new extension of -Euler numbers and polynomials related to their interpolation functions","attachmentId":78220763,"attachmentType":"pdf","work_url":"https://www.academia.edu/67385594/A_new_extension_of_Euler_numbers_and_polynomials_related_to_their_interpolation_functions","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" href="https://www.academia.edu/67385594/A_new_extension_of_Euler_numbers_and_polynomials_related_to_their_interpolation_functions"><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-related-work-sidebar-card" data-collection-position="10" data-entity-id="77100442" 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/77100442/On_Generating_Functions_for_Parametrically_Generalized_Polynomials_Involving_Combinatorial_Bernoulli_and_Euler_Polynomials_and_Numbers">On Generating Functions for Parametrically Generalized Polynomials Involving Combinatorial, Bernoulli and Euler Polynomials and Numbers</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="146832572" href="https://independent.academia.edu/YilmazSimsek2">Yilmaz Simsek</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Symmetry, 2022</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" 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style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="11" data-entity-id="65429744" 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/65429744/Combinatorial_identities_associated_with_new_families_of_the_numbers_and_polynomials_and_their_approximation_values">Combinatorial identities associated with new families of the numbers and polynomials and their approximation values</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="146832572" href="https://independent.academia.edu/YilmazSimsek2">Yilmaz Simsek</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv: Number Theory, 2017</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline 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translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="12" data-entity-id="65429687" 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/65429687/Special_Numbers_and_Polynomials_Including_Their_Generating_Functions_in_Umbral_Analysis_Methods">Special Numbers and Polynomials Including Their Generating Functions in Umbral Analysis Methods</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="146832572" href="https://independent.academia.edu/YilmazSimsek2">Yilmaz Simsek</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Axioms</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" 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js-related-work-sidebar-card" data-collection-position="13" data-entity-id="65429686" 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/65429686/Identities_Associated_with_Generalized_Stirling_Type_Numbers_and_Eulerian_Type_Polynomials">Identities Associated with Generalized Stirling Type Numbers and Eulerian Type Polynomials</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="146832572" href="https://independent.academia.edu/YilmazSimsek2">Yilmaz Simsek</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Mathematical and Computational Applications</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":"Identities Associated with Generalized Stirling Type Numbers and Eulerian Type Polynomials","attachmentId":77030806,"attachmentType":"pdf","work_url":"https://www.academia.edu/65429686/Identities_Associated_with_Generalized_Stirling_Type_Numbers_and_Eulerian_Type_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-related-work-grid-card-view-pdf" href="https://www.academia.edu/65429686/Identities_Associated_with_Generalized_Stirling_Type_Numbers_and_Eulerian_Type_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-related-work-sidebar-card" data-collection-position="14" data-entity-id="84120866" data-sort-order="default"><a class="ds-related-work--title 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M. Srivastava</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Mathematical Analysis and Applications, 2005</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":"Some generalizations of the Apostol–Bernoulli and Apostol–Euler polynomials","attachmentId":89252314,"attachmentType":"pdf","work_url":"https://www.academia.edu/84120866/Some_generalizations_of_the_Apostol_Bernoulli_and_Apostol_Euler_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-related-work-grid-card-view-pdf" href="https://www.academia.edu/84120866/Some_generalizations_of_the_Apostol_Bernoulli_and_Apostol_Euler_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-related-work-sidebar-card" data-collection-position="15" data-entity-id="120229568" 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/120229568/A_New_Family_of_Fubini_Type_Numbers_and_Polynomials_Associated_with_Apostol_Bernoulli_Numbers_and_Polynomials">A New Family of Fubini Type Numbers and Polynomials Associated with Apostol-Bernoulli Numbers and Polynomials</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="146832572" href="https://independent.academia.edu/YilmazSimsek2">Yilmaz Simsek</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of The Korean Mathematical Society, 2017</p><div 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