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window.loswp.willEdgeCache = false; window.loswp.work = {"work":{"id":15916595,"created_at":"2015-09-19T18:07:38.428-07:00","from_world_paper_id":142190594,"updated_at":"2021-01-18T10:36:53.980-08:00","_data":{"abstract":"The Stirling numbers of the first kind can be represented in terms of a new class of polynomials that are closely related to the Bernoulli polynomials. Recursion relations for these polynomials are given."},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Bernoulli-like polynomials associated with Stirling Numbers","broadcastable":true,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [35073727]; 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.loginModal = {}; window.loginModal.appleClientId = 'edu.academia.applesignon'; window.userInChina = "false";</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":41207219,"attachmentType":"pdf"}"><img alt="First page of “Bernoulli-like polynomials associated with Stirling Numbers”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/41207219/mini_magick20190219-13950-1un8kr2.png?1550630320" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/images/single_work_splash/adobe_icon.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">Bernoulli-like polynomials associated with Stirling Numbers</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="35073727" href="https://wustl.academia.edu/CarlBender"><img alt="Profile image of Carl Bender" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />Carl Bender</a></div><div class="ds-work-card--detail"><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">3 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">2 files</p></div></div><script>(async () => { const workId = 15916595; const worksViewsPath = "/v0/works/views?subdomain_param=api&work_ids%5B%5D=15916595"; 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) => { try { const viewCountNumber = parseInt(viewCount, 10); if (viewCountNumber === 0) { // Remove the whole views element if there are zero views. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); return; } const commaizedViewCount = viewCountNumber.toLocaleString(); const viewCountBody = document.getElementById('work-metadata-view-count'); if (!viewCountBody) { throw new Error('Failed to find work views element'); } viewCountBody.textContent = `${commaizedViewCount} views`; } catch (error) { // Remove the whole views element if there was some issue parsing. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); throw new Error(`Failed to parse view count: ${viewCount}`, error); } }; // 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 Stirling numbers of the first kind can be represented in terms of a new class of polynomials that are closely related to the Bernoulli polynomials. Recursion relations for these polynomials are given.</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":41207219,"attachmentType":"pdf","workUrl":"https://www.academia.edu/15916595/Bernoulli_like_polynomials_associated_with_Stirling_Numbers"}">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":41207219,"attachmentType":"pdf","workUrl":"https://www.academia.edu/15916595/Bernoulli_like_polynomials_associated_with_Stirling_Numbers"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div><div class="ds-signup-banner-trigger-container"><div class="ds-signup-banner-trigger ds-signup-banner-trigger-control"></div></div><div class="ds-signup-banner ds-signup-banner-control"><div id="ds-signup-banner-close-button"><button class="ds2-5-button ds2-5-button--secondary ds2-5-button--inverse"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">close</span></button></div><div class="ds-signup-banner-ctas" data-impression-entity-id="15916595" data-impression-entity-type="2" data-impression-source="signup-banner"><img src="//a.academia-assets.com/images/academia-logo-capital-white.svg" /><h4 class="ds2-5-heading-serif-sm">Sign up for access to the world's latest research</h4><button class="ds2-5-button ds2-5-button--inverse ds2-5-button--full-width js-swp-download-button" data-signup-modal="{"location":"signup-banner"}">Sign up for free<span class="material-symbols-outlined" style="font-size: 20px" translate="no">arrow_forward</span></button></div><div class="ds-signup-banner-divider"></div><div class="ds-signup-banner-reasons"><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Get notified about relevant papers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Save papers to use in your research</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Join the discussion with peers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Track your impact</span></div></div></div><script>(() => { // Set up signup banner show/hide behavior: // 1. 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Herein, we use the Blissard umbral approach and the familiar Bell polynomials. Links with available literature on this subject are also pointed out. The extension to the bivariate case is discussed.</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 Family of the r-Associated Stirling Numbers of the Second Kind and Generalized Bernoulli Polynomials","attachmentId":71430917,"attachmentType":"pdf","work_url":"https://www.academia.edu/55666692/A_Family_of_the_r_Associated_Stirling_Numbers_of_the_Second_Kind_and_Generalized_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 js-wsj-grid-card-view-pdf" href="https://www.academia.edu/55666692/A_Family_of_the_r_Associated_Stirling_Numbers_of_the_Second_Kind_and_Generalized_Bernoulli_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="1" data-entity-id="71818843" 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/71818843/A_new_formula_for_the_Bernoulli_numbers_of_the_second_kind_in_terms_of_the_Stirling_numbers_of_the_first_kind">A new formula for the Bernoulli numbers of the second kind in terms of the Stirling numbers of the first kind</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="5147383" href="https://henanpu.academia.edu/FengQi">Feng Qi</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Publications de l'Institut Math?matique (Belgrade)</p><p class="ds-related-work--abstract ds2-5-body-sm">We find an explicit formula for computing the Bernoulli numbers of the second kind in terms of the signed Stirling numbers of the first kind.</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 new formula for the Bernoulli numbers of the second kind in terms of the Stirling numbers of the first kind","attachmentId":81009888,"attachmentType":"pdf","work_url":"https://www.academia.edu/71818843/A_new_formula_for_the_Bernoulli_numbers_of_the_second_kind_in_terms_of_the_Stirling_numbers_of_the_first_kind","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/71818843/A_new_formula_for_the_Bernoulli_numbers_of_the_second_kind_in_terms_of_the_Stirling_numbers_of_the_first_kind"><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="2" data-entity-id="4250113" 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/4250113/Explicit_formulas_for_computing_Bernoulli_numbers_of_the_second_kind_and_Stirling_numbers_of_the_first_kind">Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind</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="5147383" href="https://henanpu.academia.edu/FengQi">Feng Qi</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2013</p><p class="ds-related-work--abstract ds2-5-body-sm">In the paper, by establishing a new and explicit formula for computing the n-th derivative of the reciprocal of the logarithmic function, the author presents new and explicit formulas for calculating Bernoulli numbers of the second kind and Stirling numbers of the first kind. As consequences of these formulas, a recursion for Stirling numbers of the first kind and a new representation of the reciprocal of the factorial n! are derived. Finally, the author finds several identities and integral representations relating to Stirling numbers of the first kind.</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 computing Bernoulli numbers of the second kind and Stirling numbers of the first kind","attachmentId":31729262,"attachmentType":"pdf","work_url":"https://www.academia.edu/4250113/Explicit_formulas_for_computing_Bernoulli_numbers_of_the_second_kind_and_Stirling_numbers_of_the_first_kind","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/4250113/Explicit_formulas_for_computing_Bernoulli_numbers_of_the_second_kind_and_Stirling_numbers_of_the_first_kind"><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="3" data-entity-id="100271655" 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/100271655/Stirlings_Series_and_Bernoulli_Numbers">Stirling's Series and Bernoulli 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="266032646" href="https://independent.academia.edu/DENNISRODRIGUEZ112">DENNIS RODRIGUEZ</a></div><p class="ds-related-work--metadata ds2-5-body-xs">The American Mathematical Monthly, 1991</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":"Stirling's Series and Bernoulli Numbers","attachmentId":101141798,"attachmentType":"pdf","work_url":"https://www.academia.edu/100271655/Stirlings_Series_and_Bernoulli_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/100271655/Stirlings_Series_and_Bernoulli_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="4" data-entity-id="56401022" 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/56401022/On_Generalized_Stirling_Numbers_and_Polynomials">On Generalized Stirling 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="12292781" href="https://independent.academia.edu/DrHariSinghParihar">Dr. Hari Singh Parihar</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2000</p><p class="ds-related-work--abstract ds2-5-body-sm">The object of this article is to present a generalization of stirling numbers and polynomials which were studied in a number of earlier work on the subject due to their importance for possible applications in certain problems arising in science and engineering (like curve fitting, coding theory, signal processing etc.). We prove that are result concerned the generalized stirling 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 Generalized Stirling Numbers and Polynomials","attachmentId":71806034,"attachmentType":"pdf","work_url":"https://www.academia.edu/56401022/On_Generalized_Stirling_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/56401022/On_Generalized_Stirling_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="5" data-entity-id="65429686" 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/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-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">Mathematical and Computational Applications</p><p class="ds-related-work--abstract ds2-5-body-sm">By using the generating functions for the generalized Stirling type numbers, Eulerian type polynomials and numbers of higher order, we derive various functional equations and differential equations. By using these equation, we derive some relations and identities related to these numbers and polynomials. Furthermore, by applying padic Volkenborn integral to these polynomials, we also derive some new identities for the generalized -Stirling type numbers of the second kind, the generalized array type polynomials and the generalized Eulerian type polynomials.</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-wsj-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-wsj-grid-card" data-collection-position="6" data-entity-id="50930552" 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/50930552/Generalized_higher_order_Stirling_numbers">Generalized higher order Stirling 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="3853163" href="https://mansoura.academia.edu/BeihElDesouky">Beih El-Desouky</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Mathematical and Computer Modelling, 2011</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper new explicit expressions for both kinds of Comtet numbers and some interesting special cases are derived. Moreover, we define and study the generalized multiparameter non-central Stirling numbers and generalized Comtet numbers via differential operators. Furthermore, recurrence relations and new explicit formulas for those numbers are obtained. Finally some interesting special cases, new combinatorial identities and a connection between these numbers and some interesting polynomials are deduced.</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":"Generalized higher order Stirling numbers","attachmentId":68807326,"attachmentType":"pdf","work_url":"https://www.academia.edu/50930552/Generalized_higher_order_Stirling_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/50930552/Generalized_higher_order_Stirling_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="63178156" 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/63178156/The_generalized_Stirling_and_Bell_numbers_revisited">The generalized Stirling and Bell numbers revisited</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="7591247" href="https://haifa.academia.edu/ToufikMansour">Toufik Mansour</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Integer Sequences, 2012</p><p class="ds-related-work--abstract ds2-5-body-sm">The generalized Stirling numbers S s;h (n, k) introduced recently by the authors are shown to be a special case of the three parameter family of generalized Stirling numbers S(n, k; α, β, r) considered by Hsu and Shiue. From this relation, several properties of S s;h (n, k) and the associated Bell numbers B s;h (n) and Bell polynomials B s;h|n (x) are derived. The particular case s = 2 and h = −1 corresponding to the meromorphic Weyl algebra is treated explicitly and its connection to Bessel numbers and Bessel polynomials is shown. The dual case s = −1 and h = 1 is connected to Hermite polynomials. For the general case, a close connection to the Touchard polynomials of higher order recently introduced by Dattoli et al. is established, and Touchard polynomials of negative order are introduced and studied. Finally, a q-analogue S s;h (n, k|q) is introduced and first properties are established, e.g., the recursion relation and an explicit expression. It is shown that the q-deformed numbers S s;h (n, k|q) are special cases of the type-II p, qanalogue of generalized Stirling numbers introduced by Remmel and Wachs, providing the analogue to the undeformed case (q = 1). Furthermore, several special cases are discussed explicitly, in particular, the case s = 2 and h = −1 corresponding to the q-meromorphic Weyl algebra considered by Diaz and Pariguan.</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":"The generalized Stirling and Bell numbers revisited","attachmentId":75687604,"attachmentType":"pdf","work_url":"https://www.academia.edu/63178156/The_generalized_Stirling_and_Bell_numbers_revisited","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/63178156/The_generalized_Stirling_and_Bell_numbers_revisited"><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="86921744" 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/86921744/Incomplete_poly_Bernoulli_numbers_associated_with_incomplete_Stirling_numbers">Incomplete poly-Bernoulli numbers associated with incomplete Stirling 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="34569352" href="https://independent.academia.edu/K%C3%A1lm%C3%A1nLiptai">Kálmán Liptai</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Publicationes Mathematicae Debrecen, 2016</p><p class="ds-related-work--abstract ds2-5-body-sm">By using the associated and restricted Stirling numbers of the second kind, we give some generalizations of the poly-Bernoulli numbers. We also study their analytic and combinatorial properties. As an application, at the end of the paper we present a new infinite series representation of the Riemann zeta function via the Lambert W .</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":"Incomplete poly-Bernoulli numbers associated with incomplete Stirling numbers","attachmentId":91268802,"attachmentType":"pdf","work_url":"https://www.academia.edu/86921744/Incomplete_poly_Bernoulli_numbers_associated_with_incomplete_Stirling_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/86921744/Incomplete_poly_Bernoulli_numbers_associated_with_incomplete_Stirling_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="9" data-entity-id="78715589" 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/78715589/_A11_Integers_13_2013_Generalized_Binomial_Expansions_and_Bernoulli_Polynomials">#A11 Integers 13 (2013) Generalized Binomial Expansions and Bernoulli 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="154361268" href="https://independent.academia.edu/HieuNguyen1425">Hieu Nguyen</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2014</p><p class="ds-related-work--abstract ds2-5-body-sm">We investigate generalized binomial expansions that arise from two-dimensional sequences satisfying a broad generalization of the triangular recurrence for binomial coefficients. 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