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The (p,q)-Stirling numbers (polynomials) of the second kind have" /> <title>(PDF) Certain Identities Associated with (p,q)-Binomial Coefficients and (p,q)-Stirling Polynomials of the Second Kind | Talha Usman - Academia.edu</title> <link rel="canonical" href="https://www.academia.edu/50434314/Certain_Identities_Associated_with_p_q_Binomial_Coefficients_and_p_q_Stirling_Polynomials_of_the_Second_Kind" /> <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 = 'c7ac46e400875c3b13c788ad246730fe5f6b36cc'; 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(1732754003000); window.Aedu.timeDifference = new Date().getTime() - 1732754003000; </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"ScholarlyArticle","abstract":"The q-Stirling numbers (polynomials) of the second kind have been investigated and applied in a variety of research subjects including, even, the q-analogue of Bernstein polynomials. The (p,q)-Stirling numbers (polynomials) of the second kind have been studied, particularly, in relation to combinatorics. In this paper, we aim to introduce new (p,q)-Stirling polynomials of the second kind which are shown to be fit for the (p,q)-analogue of Bernstein polynomials. We also present some interesting identities involving the (p,q)-binomial coefficients. We further discuss certain vanishing identities associated with the q-and (p,q)-Stirling polynomials of the second kind.","author":[{"@context":"https://schema.org","@type":"Person","name":"Talha Usman"}],"contributor":[],"dateCreated":"2021-07-31","dateModified":"2024-03-17","datePublished":null,"headline":"Certain Identities Associated with (p,q)-Binomial Coefficients and (p,q)-Stirling Polynomials of the Second Kind","inLanguage":"en","keywords":["Symmetry"],"locationCreated":null,"publication":"Symmetry","publisher":{"@context":"https://schema.org","@type":"Organization","name":"MDPI AG"},"image":null,"thumbnailUrl":null,"url":"https://www.academia.edu/50434314/Certain_Identities_Associated_with_p_q_Binomial_Coefficients_and_p_q_Stirling_Polynomials_of_the_Second_Kind","sourceOrganization":[{"@context":"https://schema.org","@type":"EducationalOrganization","name":null}]}</script><link rel="stylesheet" media="all" 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"https://www.academia.edu/login?post_login_redirect_url=https%3A%2F%2Fwww.academia.edu%2F50434314%2FCertain_Identities_Associated_with_p_q_Binomial_Coefficients_and_p_q_Stirling_Polynomials_of_the_Second_Kind%3Fauto%3Ddownload"; window.loswp.translateUrl = "https://www.academia.edu/login?post_login_redirect_url=https%3A%2F%2Fwww.academia.edu%2F50434314%2FCertain_Identities_Associated_with_p_q_Binomial_Coefficients_and_p_q_Stirling_Polynomials_of_the_Second_Kind%3Fshow_translation%3Dtrue"; window.loswp.previewableAttachments = [{"id":68424478,"identifier":"Attachment_68424478","shouldShowBulkDownload":false}]; window.loswp.shouldDetectTimezone = true; window.loswp.shouldShowBulkDownload = true; window.loswp.showSignupCaptcha = false window.loswp.willEdgeCache = false; window.loswp.work = {"work":{"id":50434314,"created_at":"2021-07-31T04:24:03.036-07:00","from_world_paper_id":171312347,"updated_at":"2024-03-15T01:05:14.467-07:00","_data":{"abstract":"The q-Stirling numbers (polynomials) of the second kind have been investigated and applied in a variety of research subjects including, even, the q-analogue of Bernstein polynomials. The (p,q)-Stirling numbers (polynomials) of the second kind have been studied, particularly, in relation to combinatorics. In this paper, we aim to introduce new (p,q)-Stirling polynomials of the second kind which are shown to be fit for the (p,q)-analogue of Bernstein polynomials. We also present some interesting identities involving the (p,q)-binomial coefficients. We further discuss certain vanishing identities associated with the q-and (p,q)-Stirling polynomials of the second kind.","publisher":"MDPI AG","publication_name":"Symmetry"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Certain Identities Associated with (p,q)-Binomial Coefficients and (p,q)-Stirling Polynomials of the Second Kind","broadcastable":true,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [71989141]; 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":68424478,"attachmentType":"pdf"}"><img alt="First page of “Certain Identities Associated with (p,q)-Binomial Coefficients and (p,q)-Stirling Polynomials of the Second Kind”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/68424478/mini_magick20210731-31531-1ncjk2k.png?1627730706" /><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">Certain Identities Associated with (p,q)-Binomial Coefficients and (p,q)-Stirling Polynomials of the Second Kind</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="71989141" href="https://independent.academia.edu/TalhaUsman5"><img alt="Profile image of Talha Usman" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />Talha Usman</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">Symmetry</p></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">The q-Stirling numbers (polynomials) of the second kind have been investigated and applied in a variety of research subjects including, even, the q-analogue of Bernstein polynomials. The (p,q)-Stirling numbers (polynomials) of the second kind have been studied, particularly, in relation to combinatorics. In this paper, we aim to introduce new (p,q)-Stirling polynomials of the second kind which are shown to be fit for the (p,q)-analogue of Bernstein polynomials. We also present some interesting identities involving the (p,q)-binomial coefficients. We further discuss certain vanishing identities associated with the q-and (p,q)-Stirling polynomials of the second kind.</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":68424478,"attachmentType":"pdf","workUrl":"https://www.academia.edu/50434314/Certain_Identities_Associated_with_p_q_Binomial_Coefficients_and_p_q_Stirling_Polynomials_of_the_Second_Kind"}">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":68424478,"attachmentType":"pdf","workUrl":"https://www.academia.edu/50434314/Certain_Identities_Associated_with_p_q_Binomial_Coefficients_and_p_q_Stirling_Polynomials_of_the_Second_Kind"}"><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="68424478" data-landing_url="https://www.academia.edu/50434314/Certain_Identities_Associated_with_p_q_Binomial_Coefficients_and_p_q_Stirling_Polynomials_of_the_Second_Kind" 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="84154891" 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/84154891/ON_q_r_w_STIRLING_NUMBERS_OF_THE_SECOND_KIND">ON (q, r, w)-STIRLING NUMBERS OF THE SECOND 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="15136743" href="https://iste-tr.academia.edu/U%C4%9EURDURAN">UĞUR DURAN</a></div><p class="ds-related-work--abstract ds2-5-body-sm">In this paper, we introduce a new generalization of the r-Stirling numbers of the second kind based on the q-numbers via an exponential generating function. 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Functions","attachmentId":44829979,"attachmentType":"pdf","work_url":"https://www.academia.edu/24497707/Generalized_q_Stirling_Numbers_and_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-wsj-grid-card-view-pdf" href="https://www.academia.edu/24497707/Generalized_q_Stirling_Numbers_and_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-wsj-grid-card" data-collection-position="7" data-entity-id="75756131" 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/75756131/The_q_log_concavity_of_the_q_Stirling_polynomials_of_the_second_kind">The q-log-concavity of the q-Stirling polynomials of the second 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="220184075" href="https://independent.academia.edu/josornonavarro">jaime andres osorno navarro</a></div><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 q-log-concavity of the q-Stirling polynomials of the second kind","attachmentId":83400430,"attachmentType":"pdf","work_url":"https://www.academia.edu/75756131/The_q_log_concavity_of_the_q_Stirling_polynomials_of_the_second_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/75756131/The_q_log_concavity_of_the_q_Stirling_polynomials_of_the_second_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="8" data-entity-id="85625135" 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/85625135/Certain_Identities_Involving_Stirling_Numbers_of_the_First_Kind">Certain Identities Involving 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="39245771" href="https://independent.academia.edu/JunesangChoi">Junesang Choi</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Far East Journal of Mathematical Sciences (FJMS), 2018</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":"Certain Identities Involving Stirling Numbers of the First Kind","attachmentId":90265050,"attachmentType":"pdf","work_url":"https://www.academia.edu/85625135/Certain_Identities_Involving_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/85625135/Certain_Identities_Involving_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="9" data-entity-id="2375290" 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/2375290/A_q_Analogue_of_the_Non_Central_Stirling_Numbers_of_the_First_Kind_and_Some_of_its_Combinatorial_Properties">A q-Analogue of the Non-Central Stirling Numbers of the First Kind and Some of its Combinatorial Properties </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="3058417" href="https://carsu.academia.edu/MiralunaHerrera">Miraluna Herrera</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Caraga State University Annals of Studies in Sciences and Humanities (a refereed journal), 2012</p><p class="ds-related-work--abstract ds2-5-body-sm">In this note, a q-analogue of the non-central Stirling numbers of the first kind is defined by means of the limit of the exponential factorial. As a consequence, some combinatorial properties, such as triangular recurrence relation, vertical recurrence relation, horizontal recurrence relation and horizontal generating function are established. 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