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backbone-social-profile-documents" style="width: 100%;"><div class="u-taCenter"></div><div class="profile--tab_content_container js-tab-pane tab-pane active" id="all"><div class="profile--tab_heading_container js-section-heading" data-section="Papers" id="Papers"><h3 class="profile--tab_heading_container">Papers by Yilmaz Simsek</h3></div><div class="js-work-strip profile--work_container" data-work-id="125444156"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125444156/Normalized_polynomials_and_their_multiplication_formulas"><img alt="Research paper thumbnail of Normalized polynomials and their multiplication formulas" class="work-thumbnail" src="https://attachments.academia-assets.com/119485941/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125444156/Normalized_polynomials_and_their_multiplication_formulas">Normalized polynomials and their multiplication formulas</a></div><div class="wp-workCard_item"><span>Advances in Difference Equations</span><span>, 2013</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="796433f9ae0e69067c8d7433951c2ef5" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119485941,"asset_id":125444156,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119485941/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span 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Type Numbers and Polynomials Associated with Apostol-Bernoulli Numbers and Polynomials" class="work-thumbnail" src="https://attachments.academia-assets.com/115448476/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" 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><div class="wp-workCard_item"><span>Journal of The Korean Mathematical Society</span><span>, 2017</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="4f4715483f76129a708a2a04bdd57d76" class="wp-workCard--action" rel="nofollow" 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We investigate some fundamental properties of these numbers and polynomials. By using generating functions and their functional equations, we derive various formulas and relations related to these numbers and polynomials. In order to compute the values of these numbers and polynomials, we give their recurrence relations. We give combinatorial sums including the Fubini type numbers and the others. 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Then one can investigate whether some generating functions can be applied to it and study what kind of new properties can be obtained by considering special generating functions. To establish that, we will use the presentations of infinite group and monoid examples, namely the split extensions Z n Z and Z 2 Z, respectively. 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data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/101102956/Derivation_of_computational_formulas_for_Changhee_polynomials_and_their_functional_and_differential_equations"><img alt="Research paper thumbnail of Derivation of computational formulas for Changhee polynomials and their functional and differential equations" class="work-thumbnail" src="https://attachments.academia-assets.com/101735105/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/101102956/Derivation_of_computational_formulas_for_Changhee_polynomials_and_their_functional_and_differential_equations">Derivation of computational formulas for Changhee polynomials and their functional and differential equations</a></div><div class="wp-workCard_item"><span>Journal of Inequalities and Applications</span><span>, 2020</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The goal of this paper is to demonstrate many explicit computational formulas and relations invol...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The goal of this paper is to demonstrate many explicit computational formulas and relations involving the Changhee polynomials and numbers and their differential equations with the help of functional equations and partial derivative equations for generating functions of these polynomials and numbers. These formulas also include the Euler polynomials, the Stirling numbers, the Bernoulli numbers and polynomials of the second kind, the Changhee polynomials of higher order, and the Daehee polynomials of higher order, which are among the well known polynomial families. By using PDEs of these generating functions, not only some recurrence relations for derivative formulas of the Changhee polynomials of higher order, but also two open problems for partial derivative equations for generating functions are given. Moreover, by using functional equations of the generating functions, two inequalities including combinatorial sums, the Changhee numbers of negative order, and the Stirling numbers ...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="fb5c91cfb2317c190fa2403d0354436e" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":101735105,"asset_id":101102956,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/101735105/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="101102956"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="101102956"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 101102956; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=101102956]").text(description); $(".js-view-count[data-work-id=101102956]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 101102956; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='101102956']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 101102956, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "fb5c91cfb2317c190fa2403d0354436e" } } $('.js-work-strip[data-work-id=101102956]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":101102956,"title":"Derivation of computational formulas for Changhee polynomials and their functional and differential equations","translated_title":"","metadata":{"abstract":"The goal of this paper is to demonstrate many explicit computational formulas and relations involving the Changhee polynomials and numbers and their differential equations with the help of functional equations and partial derivative equations for generating functions of these polynomials and numbers. These formulas also include the Euler polynomials, the Stirling numbers, the Bernoulli numbers and polynomials of the second kind, the Changhee polynomials of higher order, and the Daehee polynomials of higher order, which are among the well known polynomial families. By using PDEs of these generating functions, not only some recurrence relations for derivative formulas of the Changhee polynomials of higher order, but also two open problems for partial derivative equations for generating functions are given. Moreover, by using functional equations of the generating functions, two inequalities including combinatorial sums, the Changhee numbers of negative order, and the Stirling numbers ...","publisher":"Springer Science and Business Media LLC","publication_date":{"day":null,"month":null,"year":2020,"errors":{}},"publication_name":"Journal of Inequalities and Applications"},"translated_abstract":"The goal of this paper is to demonstrate many explicit computational formulas and relations involving the Changhee polynomials and numbers and their differential equations with the help of functional equations and partial derivative equations for generating functions of these polynomials and numbers. These formulas also include the Euler polynomials, the Stirling numbers, the Bernoulli numbers and polynomials of the second kind, the Changhee polynomials of higher order, and the Daehee polynomials of higher order, which are among the well known polynomial families. By using PDEs of these generating functions, not only some recurrence relations for derivative formulas of the Changhee polynomials of higher order, but also two open problems for partial derivative equations for generating functions are given. Moreover, by using functional equations of the generating functions, two inequalities including combinatorial sums, the Changhee numbers of negative order, and the Stirling numbers ...","internal_url":"https://www.academia.edu/101102956/Derivation_of_computational_formulas_for_Changhee_polynomials_and_their_functional_and_differential_equations","translated_internal_url":"","created_at":"2023-05-02T03:22:11.191-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":146832572,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":101735105,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/101735105/thumbnails/1.jpg","file_name":"s13660-020-02415-8.pdf","download_url":"https://www.academia.edu/attachments/101735105/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Derivation_of_computational_formulas_for.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/101735105/s13660-020-02415-8-libre.pdf?1683024465=\u0026response-content-disposition=attachment%3B+filename%3DDerivation_of_computational_formulas_for.pdf\u0026Expires=1733183070\u0026Signature=NNJ~kq9VY51pk8-BHJqNlEtEEP6jTPz51IhDFpGmwEV~AE3mEd1JLjYPwIixVnOA5fLYMXi2OtkvLfhC5VrBIT4F6Mbz1a1N-bPWWcpr9SSJIZNzuOGE4OAen3SIgQpMsZcsbdb85-0NYNOTJjqNyfN8Z5WV7wViV1cpkXSWN~soCtT~R0GpEFu27EG1Ngb-4conQs2k34CwXplXv2~sPPt5Jbc-ZxPLqxHHp02AxwvowS6AS762Be6vvW76ioxdhDFQ64nCuK4u7Bo7mjwxfFSQ-NzV7MzjQ17XEi~HGO29CFmppVAMpT6T09cXPHOpJ3KiNF-mSQ5~DXFBGNdERg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Derivation_of_computational_formulas_for_Changhee_polynomials_and_their_functional_and_differential_equations","translated_slug":"","page_count":22,"language":"en","content_type":"Work","owner":{"id":146832572,"first_name":"Yilmaz","middle_initials":null,"last_name":"Simsek","page_name":"YilmazSimsek2","domain_name":"independent","created_at":"2020-02-22T04:58:30.104-08:00","display_name":"Yilmaz Simsek","url":"https://independent.academia.edu/YilmazSimsek2"},"attachments":[{"id":101735105,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/101735105/thumbnails/1.jpg","file_name":"s13660-020-02415-8.pdf","download_url":"https://www.academia.edu/attachments/101735105/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Derivation_of_computational_formulas_for.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/101735105/s13660-020-02415-8-libre.pdf?1683024465=\u0026response-content-disposition=attachment%3B+filename%3DDerivation_of_computational_formulas_for.pdf\u0026Expires=1733183070\u0026Signature=NNJ~kq9VY51pk8-BHJqNlEtEEP6jTPz51IhDFpGmwEV~AE3mEd1JLjYPwIixVnOA5fLYMXi2OtkvLfhC5VrBIT4F6Mbz1a1N-bPWWcpr9SSJIZNzuOGE4OAen3SIgQpMsZcsbdb85-0NYNOTJjqNyfN8Z5WV7wViV1cpkXSWN~soCtT~R0GpEFu27EG1Ngb-4conQs2k34CwXplXv2~sPPt5Jbc-ZxPLqxHHp02AxwvowS6AS762Be6vvW76ioxdhDFQ64nCuK4u7Bo7mjwxfFSQ-NzV7MzjQ17XEi~HGO29CFmppVAMpT6T09cXPHOpJ3KiNF-mSQ5~DXFBGNdERg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":305,"name":"Applied Mathematics","url":"https://www.academia.edu/Documents/in/Applied_Mathematics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":207884,"name":"Recurrence Relation","url":"https://www.academia.edu/Documents/in/Recurrence_Relation"},{"id":932075,"name":"Mathematical Inequalities and Applications","url":"https://www.academia.edu/Documents/in/Mathematical_Inequalities_and_Applications"}],"urls":[{"id":31094464,"url":"http://link.springer.com/content/pdf/10.1186/s13660-020-02415-8.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="96265213"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/96265213/The_actions_on_the_generating_functions_for_the_family_of_the_generalized_Bernoulli_polynomials"><img alt="Research paper thumbnail of The actions on the generating functions for the family of the generalized Bernoulli polynomials" class="work-thumbnail" src="https://attachments.academia-assets.com/98212637/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/96265213/The_actions_on_the_generating_functions_for_the_family_of_the_generalized_Bernoulli_polynomials">The actions on the generating functions for the family of the generalized Bernoulli polynomials</a></div><div class="wp-workCard_item"><span>Filomat</span><span>, 2017</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In this paper, we study the generalization Bernoulli numbers and polynomials attached to a period...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">In this paper, we study the generalization Bernoulli numbers and polynomials attached to a periodic group homomorphism from a finite cyclic group to the set of complex numbers and derive new periodic group homomorphism by using a fixed periodic group homomorphism. Hence, we obtain not only multiplication formulas, but also some new identities for the generalized Bernoulli polynomials attached to a periodic group homomorphism.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="5f682bc9d1932d836254a49204217f7e" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":98212637,"asset_id":96265213,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/98212637/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="96265213"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="96265213"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 96265213; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=96265213]").text(description); $(".js-view-count[data-work-id=96265213]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 96265213; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='96265213']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 96265213, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "5f682bc9d1932d836254a49204217f7e" } } $('.js-work-strip[data-work-id=96265213]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":96265213,"title":"The actions on the generating functions for the family of the generalized Bernoulli polynomials","translated_title":"","metadata":{"abstract":"In this paper, we study the generalization Bernoulli numbers and polynomials attached to a periodic group homomorphism from a finite cyclic group to the set of complex numbers and derive new periodic group homomorphism by using a fixed periodic group homomorphism. Hence, we obtain not only multiplication formulas, but also some new identities for the generalized Bernoulli polynomials attached to a periodic group homomorphism.","publisher":"National Library of Serbia","publication_date":{"day":null,"month":null,"year":2017,"errors":{}},"publication_name":"Filomat"},"translated_abstract":"In this paper, we study the generalization Bernoulli numbers and polynomials attached to a periodic group homomorphism from a finite cyclic group to the set of complex numbers and derive new periodic group homomorphism by using a fixed periodic group homomorphism. Hence, we obtain not only multiplication formulas, but also some new identities for the generalized Bernoulli polynomials attached to a periodic group homomorphism.","internal_url":"https://www.academia.edu/96265213/The_actions_on_the_generating_functions_for_the_family_of_the_generalized_Bernoulli_polynomials","translated_internal_url":"","created_at":"2023-02-03T22:29:11.171-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":146832572,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":98212637,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/98212637/thumbnails/1.jpg","file_name":"61c541c2a67ae67c82ee1f2217032b43102b.pdf","download_url":"https://www.academia.edu/attachments/98212637/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"The_actions_on_the_generating_functions.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/98212637/61c541c2a67ae67c82ee1f2217032b43102b-libre.pdf?1675494178=\u0026response-content-disposition=attachment%3B+filename%3DThe_actions_on_the_generating_functions.pdf\u0026Expires=1733183070\u0026Signature=SD0OcLyf2Z~hCfkUygUcOQrcEwuX5SiBiRL9CID0efX7NyuCclIwcI3szTnKteWbUJf-OEjw51FWDH9FG1F81WBVvd45Fa2xEnAfwt0E~qIcAHPUkDPi01OrEvXnrFPDI9Nn2DxSBDPrkqhW1xkoUVhQLpheHBARIHzOfli8a1Fj2imWTG01czmkpfSdU~5NyR00f-84oZJEtQz4Qbc4grli56KAn-Oi71DQNeUTQMjJ8EmpG9YmXrDPhaCoeUztEebdGwQa7RVHh1L47F3kYmwd0S5wGVeGA3y~WXLVGjcznItlG~knEUcx4Q2O~DaMHzA74Oq7cvJv2a272TRrGQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"The_actions_on_the_generating_functions_for_the_family_of_the_generalized_Bernoulli_polynomials","translated_slug":"","page_count":10,"language":"en","content_type":"Work","owner":{"id":146832572,"first_name":"Yilmaz","middle_initials":null,"last_name":"Simsek","page_name":"YilmazSimsek2","domain_name":"independent","created_at":"2020-02-22T04:58:30.104-08:00","display_name":"Yilmaz Simsek","url":"https://independent.academia.edu/YilmazSimsek2"},"attachments":[{"id":98212637,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/98212637/thumbnails/1.jpg","file_name":"61c541c2a67ae67c82ee1f2217032b43102b.pdf","download_url":"https://www.academia.edu/attachments/98212637/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"The_actions_on_the_generating_functions.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/98212637/61c541c2a67ae67c82ee1f2217032b43102b-libre.pdf?1675494178=\u0026response-content-disposition=attachment%3B+filename%3DThe_actions_on_the_generating_functions.pdf\u0026Expires=1733183071\u0026Signature=QMn16a9Q6OBtVsehFz2Dwl2CULtawLsj0XGc5~uDw0eleOTsyjMovvPkgYuz2NwRKafkqm1TeyEJsVd7nVhlrAw9wWRYpRwqhmB9l0pgKTR7dSIRBw1HCeR5zRAC~c41oxFlLxhT1T47xXHGGUzc2bygDDmFo1PZjoUcP9kIdO01ZLhSJxk-kyX1I0~YCqQ5QmNMOhN9LIrfZuOWMzxeK9E07UEW1pewYUjZLoAongfkPGMjxMOcuwvxQbihxlFj8xS~ZgE~Q2HZSPvSLeIQgHdcDGaNWjQ06-ssFesJALUSE56kvEERseXOY3rdxcJNBBlODxAAyUOqpGCiZ97xZQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":305,"name":"Applied Mathematics","url":"https://www.academia.edu/Documents/in/Applied_Mathematics"},{"id":54284,"name":"Generalization","url":"https://www.academia.edu/Documents/in/Generalization"},{"id":3236149,"name":"Homomorphism","url":"https://www.academia.edu/Documents/in/Homomorphism"}],"urls":[]}, dispatcherData: dispatcherData }); 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Next, using the Legendre identity, trigonometric functions and the Dirichlet kernel, some formulas and relations involving Bernoulli and Euler numbers, cosine-type Bernoulli and Euler polynomials, and sine-type Bernoulli and Euler polynomials are driven. Then, by using the generating functions method and the well-known Euler identity, many new identities, formulas, and combinatorial sums among the Fibonacci numbers and polynomials, the Lucas numbers and polynomials, the Chebyshev polynomials, and Bernoulli and Euler type polynomials are given. Finally, some infinite series representations for these special numbers and polynomials and their numerical examples are presented.","publication_date":{"day":null,"month":null,"year":2020,"errors":{}},"publication_name":"TURKISH JOURNAL OF MATHEMATICS","grobid_abstract_attachment_id":93611581},"translated_abstract":null,"internal_url":"https://www.academia.edu/89898484/A_special_approach_to_derive_new_formulas_for_some_special_numbers_and_polynomials","translated_internal_url":"","created_at":"2022-11-03T12:28:10.428-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":146832572,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":93611581,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/93611581/thumbnails/1.jpg","file_name":"openAcceptedDocument.pdf","download_url":"https://www.academia.edu/attachments/93611581/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"A_special_approach_to_derive_new_formula.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/93611581/openAcceptedDocument-libre.pdf?1667506642=\u0026response-content-disposition=attachment%3B+filename%3DA_special_approach_to_derive_new_formula.pdf\u0026Expires=1733183071\u0026Signature=Mm5HprYrHEwbH379GpKngqR0XK37Dv1PH7THhd3UoW8J33~7Nf9MI~u0mumk1zmiSC77oOlGlCLQ7L7WH7~A91LvaIB2dO9zrGIK-dNf92tl8~BcBNNf7yoLMKeWAdZ46no0G3ko92rjxyjjJjUGNxgl7UEX0-YBeDSDY3VfIFRBER12dzcs582hRluzI~~g7EVP0Ei5X~xNMJZTloMgldfDCrHBltTSLHxgN54Qr3JdEYgZrKOJz-RGG9R8tXhxZqTF6d9HZxfuycWtyWtTVVsH305PTtYyDbV2j3-~xXWdjrlXzim2M5BPfKHEQNIj6-nn5keD2HJNnI8Zh0AXUQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"A_special_approach_to_derive_new_formulas_for_some_special_numbers_and_polynomials","translated_slug":"","page_count":24,"language":"en","content_type":"Work","owner":{"id":146832572,"first_name":"Yilmaz","middle_initials":null,"last_name":"Simsek","page_name":"YilmazSimsek2","domain_name":"independent","created_at":"2020-02-22T04:58:30.104-08:00","display_name":"Yilmaz Simsek","url":"https://independent.academia.edu/YilmazSimsek2"},"attachments":[{"id":93611581,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/93611581/thumbnails/1.jpg","file_name":"openAcceptedDocument.pdf","download_url":"https://www.academia.edu/attachments/93611581/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"A_special_approach_to_derive_new_formula.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/93611581/openAcceptedDocument-libre.pdf?1667506642=\u0026response-content-disposition=attachment%3B+filename%3DA_special_approach_to_derive_new_formula.pdf\u0026Expires=1733183071\u0026Signature=Mm5HprYrHEwbH379GpKngqR0XK37Dv1PH7THhd3UoW8J33~7Nf9MI~u0mumk1zmiSC77oOlGlCLQ7L7WH7~A91LvaIB2dO9zrGIK-dNf92tl8~BcBNNf7yoLMKeWAdZ46no0G3ko92rjxyjjJjUGNxgl7UEX0-YBeDSDY3VfIFRBER12dzcs582hRluzI~~g7EVP0Ei5X~xNMJZTloMgldfDCrHBltTSLHxgN54Qr3JdEYgZrKOJz-RGG9R8tXhxZqTF6d9HZxfuycWtyWtTVVsH305PTtYyDbV2j3-~xXWdjrlXzim2M5BPfKHEQNIj6-nn5keD2HJNnI8Zh0AXUQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="81374464"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/81374464/Identities_for_Dirichlet_and_Lambert_type_series_arising_from_the_numbers_of_a_certain_special_word"><img alt="Research paper thumbnail of Identities for Dirichlet and Lambert-type series arising from the numbers of a certain special word" class="work-thumbnail" src="https://attachments.academia-assets.com/87438925/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/81374464/Identities_for_Dirichlet_and_Lambert_type_series_arising_from_the_numbers_of_a_certain_special_word">Identities for Dirichlet and Lambert-type series arising from the numbers of a certain special word</a></div><div class="wp-workCard_item"><span>Applicable Analysis and Discrete Mathematics</span><span>, 2019</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The goal of this paper is to give several new Dirichlet-type series associated with the Riemann z...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The goal of this paper is to give several new Dirichlet-type series associated with the Riemann zeta function, the polylogarithm function, and also the numbers of necklaces and Lyndon words. By applying Dirichlet convolution formula to number-theoretic functions related to these series, various novel identities and relations are derived. Moreover, some new formulas related to Bernoulli-type numbers and polynomials obtain from generating functions and these Dirichlet-type series. Finally, several relations among the Fourier expansion of Eisenstein series, the Lambert series and the number-theoretic functions are given.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="0a39c850afe2440d093988181f072683" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":87438925,"asset_id":81374464,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/87438925/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="81374464"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="81374464"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 81374464; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=81374464]").text(description); $(".js-view-count[data-work-id=81374464]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 81374464; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='81374464']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 81374464, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "0a39c850afe2440d093988181f072683" } } $('.js-work-strip[data-work-id=81374464]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":81374464,"title":"Identities for Dirichlet and Lambert-type series arising from the numbers of a certain special word","translated_title":"","metadata":{"abstract":"The goal of this paper is to give several new Dirichlet-type series associated with the Riemann zeta function, the polylogarithm function, and also the numbers of necklaces and Lyndon words. By applying Dirichlet convolution formula to number-theoretic functions related to these series, various novel identities and relations are derived. Moreover, some new formulas related to Bernoulli-type numbers and polynomials obtain from generating functions and these Dirichlet-type series. Finally, several relations among the Fourier expansion of Eisenstein series, the Lambert series and the number-theoretic functions are given.","publisher":"National Library of Serbia","publication_date":{"day":null,"month":null,"year":2019,"errors":{}},"publication_name":"Applicable Analysis and Discrete Mathematics"},"translated_abstract":"The goal of this paper is to give several new Dirichlet-type series associated with the Riemann zeta function, the polylogarithm function, and also the numbers of necklaces and Lyndon words. By applying Dirichlet convolution formula to number-theoretic functions related to these series, various novel identities and relations are derived. Moreover, some new formulas related to Bernoulli-type numbers and polynomials obtain from generating functions and these Dirichlet-type series. Finally, several relations among the Fourier expansion of Eisenstein series, the Lambert series and the number-theoretic functions are given.","internal_url":"https://www.academia.edu/81374464/Identities_for_Dirichlet_and_Lambert_type_series_arising_from_the_numbers_of_a_certain_special_word","translated_internal_url":"","created_at":"2022-06-13T04:03:09.837-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":146832572,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":87438925,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/87438925/thumbnails/1.jpg","file_name":"1452-86301900033K.pdf","download_url":"https://www.academia.edu/attachments/87438925/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Identities_for_Dirichlet_and_Lambert_typ.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/87438925/1452-86301900033K-libre.pdf?1655118710=\u0026response-content-disposition=attachment%3B+filename%3DIdentities_for_Dirichlet_and_Lambert_typ.pdf\u0026Expires=1733183071\u0026Signature=Vr4DEmX6QRJxLLQGzxPFA0uBTkTHir4PP1LnMEFxxHjukQptPdC1vATTI0f1-5EzQlTKgIblp6Qbp5m58szWvrolyMOsimnkkiZiI5gFdyYa5d5cshpmrpZdL-2y51qp6cvz7O9bDgvcV8H1o2A7ItccrOTIhWQUowI-gKSVSv9ENuMOW~VJIJIyn-p0nvwrIzqOA-8d1GTUKXSvKnjLvKLM39jYSDMsEnTDnR24UZl6aYdY3LH8vag9O4Y9VNVcRGlZwU2G7XVjczb6xyEeIRzy3i877-AyJxrjskLGcx11iLrsl2u-kgBCBRL0N5u4DDaS7~H4yEWgQJzDLkcfcA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Identities_for_Dirichlet_and_Lambert_type_series_arising_from_the_numbers_of_a_certain_special_word","translated_slug":"","page_count":18,"language":"en","content_type":"Work","owner":{"id":146832572,"first_name":"Yilmaz","middle_initials":null,"last_name":"Simsek","page_name":"YilmazSimsek2","domain_name":"independent","created_at":"2020-02-22T04:58:30.104-08:00","display_name":"Yilmaz Simsek","url":"https://independent.academia.edu/YilmazSimsek2"},"attachments":[{"id":87438925,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/87438925/thumbnails/1.jpg","file_name":"1452-86301900033K.pdf","download_url":"https://www.academia.edu/attachments/87438925/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Identities_for_Dirichlet_and_Lambert_typ.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/87438925/1452-86301900033K-libre.pdf?1655118710=\u0026response-content-disposition=attachment%3B+filename%3DIdentities_for_Dirichlet_and_Lambert_typ.pdf\u0026Expires=1733183071\u0026Signature=Vr4DEmX6QRJxLLQGzxPFA0uBTkTHir4PP1LnMEFxxHjukQptPdC1vATTI0f1-5EzQlTKgIblp6Qbp5m58szWvrolyMOsimnkkiZiI5gFdyYa5d5cshpmrpZdL-2y51qp6cvz7O9bDgvcV8H1o2A7ItccrOTIhWQUowI-gKSVSv9ENuMOW~VJIJIyn-p0nvwrIzqOA-8d1GTUKXSvKnjLvKLM39jYSDMsEnTDnR24UZl6aYdY3LH8vag9O4Y9VNVcRGlZwU2G7XVjczb6xyEeIRzy3i877-AyJxrjskLGcx11iLrsl2u-kgBCBRL0N5u4DDaS7~H4yEWgQJzDLkcfcA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":556845,"name":"Numerical Analysis and Computational Mathematics","url":"https://www.academia.edu/Documents/in/Numerical_Analysis_and_Computational_Mathematics"},{"id":3777346,"name":"Dirichlet series","url":"https://www.academia.edu/Documents/in/Dirichlet_series"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="77100443"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" rel="nofollow" href="https://www.academia.edu/77100443/Some_Certain_Classes_of_Combinatorial_Numbers_and_Polynomials_Attached_to_Dirichlet_Characters_Their_Construction_by_p_Adic_Integration_and_Applications_to_Probability_Distribution_Functions"><img alt="Research paper thumbnail of Some Certain Classes of Combinatorial Numbers and Polynomials Attached to Dirichlet Characters: Their Construction by p-Adic Integration and Applications to Probability Distribution Functions" class="work-thumbnail" src="https://a.academia-assets.com/images/blank-paper.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" rel="nofollow" href="https://www.academia.edu/77100443/Some_Certain_Classes_of_Combinatorial_Numbers_and_Polynomials_Attached_to_Dirichlet_Characters_Their_Construction_by_p_Adic_Integration_and_Applications_to_Probability_Distribution_Functions">Some Certain Classes of Combinatorial Numbers and Polynomials Attached to Dirichlet Characters: Their Construction by p-Adic Integration and Applications to Probability Distribution Functions</a></div><div class="wp-workCard_item"><span>Mathematical Analysis in Interdisciplinary Research</span><span>, 2012</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77100443"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77100443"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77100443; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77100443]").text(description); $(".js-view-count[data-work-id=77100443]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77100443; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77100443']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77100443, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (false){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="77100442"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/77100442/On_Generating_Functions_for_Parametrically_Generalized_Polynomials_Involving_Combinatorial_Bernoulli_and_Euler_Polynomials_and_Numbers"><img alt="Research paper thumbnail of On Generating Functions for Parametrically Generalized Polynomials Involving Combinatorial, Bernoulli and Euler Polynomials and Numbers" class="work-thumbnail" src="https://attachments.academia-assets.com/84559082/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" 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><div class="wp-workCard_item"><span>Symmetry</span><span>, 2022</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The aim of this paper is to give generating functions for parametrically generalized polynomials ...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The aim of this paper is to give generating functions for parametrically generalized polynomials that are related to the combinatorial numbers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the cosine-Bernoulli polynomials, the sine-Bernoulli polynomials, the cosine-Euler polynomials, and the sine-Euler polynomials. We investigate some properties of these generating functions. By applying Euler’s formula to these generating functions, we derive many new and interesting formulas and relations related to these special polynomials and numbers mentioned as above. Some special cases of the results obtained in this article are examined. With this special case, detailed comments and comparisons with previously available results are also provided. Furthermore, we come up with open questions about interpolation functions for these polynomials. The main results of this paper highlight the existing symmetry between numbers and polynomials in a more general framework...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="0d037e844b355f646fe9070914d6c26a" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":84559082,"asset_id":77100442,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/84559082/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77100442"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77100442"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77100442; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77100442]").text(description); $(".js-view-count[data-work-id=77100442]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77100442; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77100442']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77100442, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "0d037e844b355f646fe9070914d6c26a" } } $('.js-work-strip[data-work-id=77100442]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77100442,"title":"On Generating Functions for Parametrically Generalized Polynomials Involving Combinatorial, Bernoulli and Euler Polynomials and Numbers","translated_title":"","metadata":{"abstract":"The aim of this paper is to give generating functions for parametrically generalized polynomials that are related to the combinatorial numbers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the cosine-Bernoulli polynomials, the sine-Bernoulli polynomials, the cosine-Euler polynomials, and the sine-Euler polynomials. We investigate some properties of these generating functions. By applying Euler’s formula to these generating functions, we derive many new and interesting formulas and relations related to these special polynomials and numbers mentioned as above. Some special cases of the results obtained in this article are examined. With this special case, detailed comments and comparisons with previously available results are also provided. Furthermore, we come up with open questions about interpolation functions for these polynomials. The main results of this paper highlight the existing symmetry between numbers and polynomials in a more general framework...","publisher":"MDPI AG","publication_date":{"day":null,"month":null,"year":2022,"errors":{}},"publication_name":"Symmetry"},"translated_abstract":"The aim of this paper is to give generating functions for parametrically generalized polynomials that are related to the combinatorial numbers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the cosine-Bernoulli polynomials, the sine-Bernoulli polynomials, the cosine-Euler polynomials, and the sine-Euler polynomials. We investigate some properties of these generating functions. By applying Euler’s formula to these generating functions, we derive many new and interesting formulas and relations related to these special polynomials and numbers mentioned as above. Some special cases of the results obtained in this article are examined. With this special case, detailed comments and comparisons with previously available results are also provided. Furthermore, we come up with open questions about interpolation functions for these polynomials. The main results of this paper highlight the existing symmetry between numbers and polynomials in a more general framework...","internal_url":"https://www.academia.edu/77100442/On_Generating_Functions_for_Parametrically_Generalized_Polynomials_Involving_Combinatorial_Bernoulli_and_Euler_Polynomials_and_Numbers","translated_internal_url":"","created_at":"2022-04-20T11:12:48.092-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":146832572,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":84559082,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559082/thumbnails/1.jpg","file_name":"pdf.pdf","download_url":"https://www.academia.edu/attachments/84559082/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_Generating_Functions_for_Parametrical.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559082/pdf-libre.pdf?1650478773=\u0026response-content-disposition=attachment%3B+filename%3DOn_Generating_Functions_for_Parametrical.pdf\u0026Expires=1733183071\u0026Signature=XAnw-zkJfw~6OB17TCkeDY7hdV3KtfVAuiCDJd~wSzfHnTnUn3d5wtyaCLXMJFzU~qsiSXeS6vMM-WTOlMON2QegbRVqQFeN4F0LL9qAD0AtPvPmQOzPY~hukFmGqpPH7-2HWpJc8auCZgE0e73v-4L9LdhNVcr~wGZg0Vgjbkrys6OwH2NysEWRYoZIuOJprE-0sgtcXxlBTRMZHyLUVXabYEB0bnej3nbQ2wRDwd~07Joh1qk6clx07RFDx9sgHiZBjCyreze1jm2Ju7DEo-2mDp-IFYtA5cmB8NYRepj1baIGPraSy~O6W6D--BrBJyB2DZD8qRa5W4ibD8MvPA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"On_Generating_Functions_for_Parametrically_Generalized_Polynomials_Involving_Combinatorial_Bernoulli_and_Euler_Polynomials_and_Numbers","translated_slug":"","page_count":12,"language":"en","content_type":"Work","owner":{"id":146832572,"first_name":"Yilmaz","middle_initials":null,"last_name":"Simsek","page_name":"YilmazSimsek2","domain_name":"independent","created_at":"2020-02-22T04:58:30.104-08:00","display_name":"Yilmaz Simsek","url":"https://independent.academia.edu/YilmazSimsek2"},"attachments":[{"id":84559082,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559082/thumbnails/1.jpg","file_name":"pdf.pdf","download_url":"https://www.academia.edu/attachments/84559082/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_Generating_Functions_for_Parametrical.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559082/pdf-libre.pdf?1650478773=\u0026response-content-disposition=attachment%3B+filename%3DOn_Generating_Functions_for_Parametrical.pdf\u0026Expires=1733183071\u0026Signature=XAnw-zkJfw~6OB17TCkeDY7hdV3KtfVAuiCDJd~wSzfHnTnUn3d5wtyaCLXMJFzU~qsiSXeS6vMM-WTOlMON2QegbRVqQFeN4F0LL9qAD0AtPvPmQOzPY~hukFmGqpPH7-2HWpJc8auCZgE0e73v-4L9LdhNVcr~wGZg0Vgjbkrys6OwH2NysEWRYoZIuOJprE-0sgtcXxlBTRMZHyLUVXabYEB0bnej3nbQ2wRDwd~07Joh1qk6clx07RFDx9sgHiZBjCyreze1jm2Ju7DEo-2mDp-IFYtA5cmB8NYRepj1baIGPraSy~O6W6D--BrBJyB2DZD8qRa5W4ibD8MvPA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":126829,"name":"Symmetry","url":"https://www.academia.edu/Documents/in/Symmetry"}],"urls":[{"id":19721510,"url":"https://www.mdpi.com/2073-8994/14/4/654/pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="77100441"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" rel="nofollow" href="https://www.academia.edu/77100441/Formulas_for_characteristic_function_and_moment_generating_functions_of_beta_type_distribution"><img alt="Research paper thumbnail of Formulas for characteristic function and moment generating functions of beta type distribution" class="work-thumbnail" src="https://a.academia-assets.com/images/blank-paper.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" rel="nofollow" href="https://www.academia.edu/77100441/Formulas_for_characteristic_function_and_moment_generating_functions_of_beta_type_distribution">Formulas for characteristic function and moment generating functions of beta type distribution</a></div><div class="wp-workCard_item"><span>Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas</span><span>, 2022</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77100441"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77100441"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77100441; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77100441]").text(description); $(".js-view-count[data-work-id=77100441]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77100441; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77100441']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77100441, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (false){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "-1" } } $('.js-work-strip[data-work-id=77100441]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77100441,"title":"Formulas for characteristic function and moment generating functions of beta type distribution","translated_title":"","metadata":{"publisher":"Springer Science and Business Media LLC","publication_date":{"day":null,"month":null,"year":2022,"errors":{}},"publication_name":"Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. 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Reciprocity laws for these sums are proven. Some applications of these reciprocity laws are presented. With aid of the reciprocity law of the Dedekind sums, formulas for many new finite sums are obtained. Relations among these new classes of finite sums, partial sum of the generalized harmonic functions, the Riemann zeta function, the Hurwitz zeta function, hypergeometric series, polylogarithms, digamma functions, polygmamma functions, and special numbers and polynomials and other combinatorial sums are given. Moreover, some formulas for the partial sum of the generalized harmonic functions and special numbers and polynomials are given. Finally, coments and observations on the results of this paper are given.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2b2527cedbf392f40d931058f88f1e08" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":84559080,"asset_id":77100440,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/84559080/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77100440"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77100440"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77100440; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77100440]").text(description); $(".js-view-count[data-work-id=77100440]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77100440; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77100440']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77100440, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "2b2527cedbf392f40d931058f88f1e08" } } $('.js-work-strip[data-work-id=77100440]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77100440,"title":"Some Classes of Finite Sums Related to the Generalized Harmonic Functions and Special Numbers and Polynomials","translated_title":"","metadata":{"abstract":"The aim of this paper is to give some new classes of finite sums involving the numbers y (m, λ), the generalized harmonic functions, special numbers and polynomials, the Dedekind sums, and other combinatorial sum. Reciprocity laws for these sums are proven. Some applications of these reciprocity laws are presented. With aid of the reciprocity law of the Dedekind sums, formulas for many new finite sums are obtained. Relations among these new classes of finite sums, partial sum of the generalized harmonic functions, the Riemann zeta function, the Hurwitz zeta function, hypergeometric series, polylogarithms, digamma functions, polygmamma functions, and special numbers and polynomials and other combinatorial sums are given. Moreover, some formulas for the partial sum of the generalized harmonic functions and special numbers and polynomials are given. Finally, coments and observations on the results of this paper are given.","publication_date":{"day":null,"month":null,"year":2022,"errors":{}}},"translated_abstract":"The aim of this paper is to give some new classes of finite sums involving the numbers y (m, λ), the generalized harmonic functions, special numbers and polynomials, the Dedekind sums, and other combinatorial sum. Reciprocity laws for these sums are proven. Some applications of these reciprocity laws are presented. With aid of the reciprocity law of the Dedekind sums, formulas for many new finite sums are obtained. Relations among these new classes of finite sums, partial sum of the generalized harmonic functions, the Riemann zeta function, the Hurwitz zeta function, hypergeometric series, polylogarithms, digamma functions, polygmamma functions, and special numbers and polynomials and other combinatorial sums are given. Moreover, some formulas for the partial sum of the generalized harmonic functions and special numbers and polynomials are given. Finally, coments and observations on the results of this paper are given.","internal_url":"https://www.academia.edu/77100440/Some_Classes_of_Finite_Sums_Related_to_the_Generalized_Harmonic_Functions_and_Special_Numbers_and_Polynomials","translated_internal_url":"","created_at":"2022-04-20T11:12:47.638-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":146832572,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":84559080,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559080/thumbnails/1.jpg","file_name":"MTJPAM-D-21-00002.pdf","download_url":"https://www.academia.edu/attachments/84559080/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Some_Classes_of_Finite_Sums_Related_to_t.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559080/MTJPAM-D-21-00002-libre.pdf?1650478773=\u0026response-content-disposition=attachment%3B+filename%3DSome_Classes_of_Finite_Sums_Related_to_t.pdf\u0026Expires=1733183071\u0026Signature=Bj8kwiIhEhT3ffz0s5q7ZMgujR4l8pT5rDegOHtapw44nOSnLZafHz31LKlz40DfTZOq0HLKIx~76G3UYMJk-foKwKtPyuDHLxV5bPzA-PeR9G85HfjlH1PMAXmuMkNzTOoIUS3aPgeX4N18emQyzqRaiG4-IOJgvS7-VurdizJW52DyKoc4ep47a~PVVjrf9czUyJmRwynq1ZQIFTU5yqBqJPbXupuzv7GpFb5y555J0mkembTjv1kkB2cPmSMaryi~mgcgsGpPotr~TeAnf7xE~tEmO6KZlq0BakFTQJo-BrAc16X1B8MfQ2rLyrK-CJQhbW0iTFoTZHSoWM8cKA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Some_Classes_of_Finite_Sums_Related_to_the_Generalized_Harmonic_Functions_and_Special_Numbers_and_Polynomials","translated_slug":"","page_count":19,"language":"en","content_type":"Work","owner":{"id":146832572,"first_name":"Yilmaz","middle_initials":null,"last_name":"Simsek","page_name":"YilmazSimsek2","domain_name":"independent","created_at":"2020-02-22T04:58:30.104-08:00","display_name":"Yilmaz Simsek","url":"https://independent.academia.edu/YilmazSimsek2"},"attachments":[{"id":84559080,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559080/thumbnails/1.jpg","file_name":"MTJPAM-D-21-00002.pdf","download_url":"https://www.academia.edu/attachments/84559080/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Some_Classes_of_Finite_Sums_Related_to_t.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559080/MTJPAM-D-21-00002-libre.pdf?1650478773=\u0026response-content-disposition=attachment%3B+filename%3DSome_Classes_of_Finite_Sums_Related_to_t.pdf\u0026Expires=1733183071\u0026Signature=Bj8kwiIhEhT3ffz0s5q7ZMgujR4l8pT5rDegOHtapw44nOSnLZafHz31LKlz40DfTZOq0HLKIx~76G3UYMJk-foKwKtPyuDHLxV5bPzA-PeR9G85HfjlH1PMAXmuMkNzTOoIUS3aPgeX4N18emQyzqRaiG4-IOJgvS7-VurdizJW52DyKoc4ep47a~PVVjrf9czUyJmRwynq1ZQIFTU5yqBqJPbXupuzv7GpFb5y555J0mkembTjv1kkB2cPmSMaryi~mgcgsGpPotr~TeAnf7xE~tEmO6KZlq0BakFTQJo-BrAc16X1B8MfQ2rLyrK-CJQhbW0iTFoTZHSoWM8cKA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":84559081,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559081/thumbnails/1.jpg","file_name":"MTJPAM-D-21-00002.pdf","download_url":"https://www.academia.edu/attachments/84559081/download_file","bulk_download_file_name":"Some_Classes_of_Finite_Sums_Related_to_t.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559081/MTJPAM-D-21-00002-libre.pdf?1650478774=\u0026response-content-disposition=attachment%3B+filename%3DSome_Classes_of_Finite_Sums_Related_to_t.pdf\u0026Expires=1733183071\u0026Signature=K2PPLuVPwQWAHpa3o1np0fs5iSiL9GPAd3v2i5kcV0esuLvcKPrkyNv891iZrf71kmGrVKhQfpvcEFVcW~MUHP9n1phXD7K3v-F0f-~ijHLiFlshuS7MbC2puYSqGwzi07dA6FFUO5QAaQp0m4YUhaCZTENDlfmPeAJZo7-yyON7seajnmSVeFiG9yVWaQ3-Eip8jo~WX~Xml3NqfQOf5UyFn6rqYGnqoTQmbi4EXmrZfD1~v~SlG42YKJfoXbOfItgyyo9JLql3d1VUJ2l5GBKMaMO12ziDdR874xjryZasBodW7-ffRKuU1a5DhGE1puzPznbDe7S5GIaBqjCFYA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[],"urls":[{"id":19721508,"url":"https://mtjpamjournal.com/wp-content/uploads/2021/10/MTJPAM-D-21-00002.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="77100439"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/77100439/Computation_Methods_for_combinatorial_sums_and_Euler_type_numbers_related_to_new_families_of_numbers"><img alt="Research paper thumbnail of Computation Methods for combinatorial sums and Euler type numbers related to new families of numbers" class="work-thumbnail" src="https://attachments.academia-assets.com/84559078/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/77100439/Computation_Methods_for_combinatorial_sums_and_Euler_type_numbers_related_to_new_families_of_numbers">Computation Methods for combinatorial sums and Euler type numbers related to new families of numbers</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The aim of this article is to define some new families of the special numbers. These numbers prov...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order. We can show that these numbers are related to the well-known numbers and polynomials such as the Stirling numbers of the second kind and the central factorial numbers, the array polynomials, the rook numbers and polynomials, the Bernstein basis functions and others. In order to derive our new identities and relations for these numbers, we use a technique including the generating functions and functional equations. Finally, we not only give a computational algorithm for these numbers, but also some numerical values of these numbers and the Euler numbers of negative order with tables. We also give some combinatorial interpretations of our new numbers.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="8bc2052e03bc43a6c807b49282448e3f" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":84559078,"asset_id":77100439,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/84559078/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77100439"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77100439"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77100439; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77100439]").text(description); $(".js-view-count[data-work-id=77100439]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77100439; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77100439']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77100439, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "8bc2052e03bc43a6c807b49282448e3f" } } $('.js-work-strip[data-work-id=77100439]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77100439,"title":"Computation Methods for combinatorial sums and Euler type numbers related to new families of numbers","translated_title":"","metadata":{"abstract":"The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order. We can show that these numbers are related to the well-known numbers and polynomials such as the Stirling numbers of the second kind and the central factorial numbers, the array polynomials, the rook numbers and polynomials, the Bernstein basis functions and others. In order to derive our new identities and relations for these numbers, we use a technique including the generating functions and functional equations. Finally, we not only give a computational algorithm for these numbers, but also some numerical values of these numbers and the Euler numbers of negative order with tables. We also give some combinatorial interpretations of our new numbers.","ai_title_tag":"New Families of Special Numbers and Their Combinatorial Sums","publication_date":{"day":19,"month":4,"year":2016,"errors":{}}},"translated_abstract":"The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order. We can show that these numbers are related to the well-known numbers and polynomials such as the Stirling numbers of the second kind and the central factorial numbers, the array polynomials, the rook numbers and polynomials, the Bernstein basis functions and others. In order to derive our new identities and relations for these numbers, we use a technique including the generating functions and functional equations. Finally, we not only give a computational algorithm for these numbers, but also some numerical values of these numbers and the Euler numbers of negative order with tables. We also give some combinatorial interpretations of our new numbers.","internal_url":"https://www.academia.edu/77100439/Computation_Methods_for_combinatorial_sums_and_Euler_type_numbers_related_to_new_families_of_numbers","translated_internal_url":"","created_at":"2022-04-20T11:12:47.363-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":146832572,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":84559078,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559078/thumbnails/1.jpg","file_name":"1604.05608v1.pdf","download_url":"https://www.academia.edu/attachments/84559078/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Computation_Methods_for_combinatorial_su.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559078/1604.05608v1-libre.pdf?1650478774=\u0026response-content-disposition=attachment%3B+filename%3DComputation_Methods_for_combinatorial_su.pdf\u0026Expires=1733183071\u0026Signature=GZBWp2sLrVc8PABPmSiX763uCjmVF28j-gjtYHzifEkzK99RJHYng16NZopnmziGddL56K-JzKgUvRN6WfgdfLysaCL5GwN3SFm5upCHOeo-YEySIHuKgp2csg4p3Zk1EcYgOwul2hyKa6-sSysQKh-l2UuS0fAP0IpUQHXmAP~pIlpdKcCkW7NAMFcw4zCdHdkl3-cjI3MhfdruXK-unfUuZ2PDOQg8PuCCvfwnS5HCN-qmxenAHAYrGCOpcgfVekGSgr~Z2jpggMkPvGlxq2ofXYNpxoushL6j5Y8jQekR~qDJPe7zIsNh7WGlplJ66ypYb8AfCHLgHuYAcdDonQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Computation_Methods_for_combinatorial_sums_and_Euler_type_numbers_related_to_new_families_of_numbers","translated_slug":"","page_count":19,"language":"en","content_type":"Work","owner":{"id":146832572,"first_name":"Yilmaz","middle_initials":null,"last_name":"Simsek","page_name":"YilmazSimsek2","domain_name":"independent","created_at":"2020-02-22T04:58:30.104-08:00","display_name":"Yilmaz Simsek","url":"https://independent.academia.edu/YilmazSimsek2"},"attachments":[{"id":84559078,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559078/thumbnails/1.jpg","file_name":"1604.05608v1.pdf","download_url":"https://www.academia.edu/attachments/84559078/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Computation_Methods_for_combinatorial_su.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559078/1604.05608v1-libre.pdf?1650478774=\u0026response-content-disposition=attachment%3B+filename%3DComputation_Methods_for_combinatorial_su.pdf\u0026Expires=1733183071\u0026Signature=GZBWp2sLrVc8PABPmSiX763uCjmVF28j-gjtYHzifEkzK99RJHYng16NZopnmziGddL56K-JzKgUvRN6WfgdfLysaCL5GwN3SFm5upCHOeo-YEySIHuKgp2csg4p3Zk1EcYgOwul2hyKa6-sSysQKh-l2UuS0fAP0IpUQHXmAP~pIlpdKcCkW7NAMFcw4zCdHdkl3-cjI3MhfdruXK-unfUuZ2PDOQg8PuCCvfwnS5HCN-qmxenAHAYrGCOpcgfVekGSgr~Z2jpggMkPvGlxq2ofXYNpxoushL6j5Y8jQekR~qDJPe7zIsNh7WGlplJ66ypYb8AfCHLgHuYAcdDonQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":84559079,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559079/thumbnails/1.jpg","file_name":"1604.05608v1.pdf","download_url":"https://www.academia.edu/attachments/84559079/download_file","bulk_download_file_name":"Computation_Methods_for_combinatorial_su.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559079/1604.05608v1-libre.pdf?1650478773=\u0026response-content-disposition=attachment%3B+filename%3DComputation_Methods_for_combinatorial_su.pdf\u0026Expires=1733183071\u0026Signature=E64c~os-IuQ2ekKJdYgnMKDTz-WRyXe4qFdz5uYOGdQpnedh6JK3lQN~BM1XpL24dLiGd43NQBQ5ZCBS5uBzJdqfYZrB9uqg6SAvwT5XAo9GLn~nnZ6GJBXK1-vpSvno9TO-OArTULbcm9f0rlP9BmqH0B1~uPAqKzOPtbl3hYJCdCKwEz~68THbIpYvxY1rMc8j80XuzVX~HFC6Kk~1E9~WlqTuBaK6bR~A5L-Y-PfUaeBKTgdR3r~nc2QS9KzEFWMUXvIP08~dgNzHNwDhG5DCdMu1FToVfnlyMDkUJDIephj4tIrfWOBZgXcoZwIZ63B8AhqXZCuQY0C3tS0CYg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":305,"name":"Applied Mathematics","url":"https://www.academia.edu/Documents/in/Applied_Mathematics"}],"urls":[{"id":19721507,"url":"https://arxiv.org/pdf/1604.05608v1.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="77100438"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/77100438/New_families_of_special_numbers_for_computing_negative_order_Euler_numbers"><img alt="Research paper thumbnail of New families of special numbers for computing negative order Euler numbers" class="work-thumbnail" src="https://attachments.academia-assets.com/84559077/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/77100438/New_families_of_special_numbers_for_computing_negative_order_Euler_numbers">New families of special numbers for computing negative order Euler numbers</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The main purpose of this paper is to construct new families of special numbers with their generat...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The main purpose of this paper is to construct new families of special numbers with their generating functions. These numbers are related to the many well-known numbers, which are the Bernoulli numbers, the Fibonacci numbers, the Lucas numbers, the Stirling numbers of the second kind and the central factorial numbers. Our other inspiration of this paper is related to the Golombek&#39;s problem golombek“ Aufgabe 1088, El. Math. 49 (1994) 126-127” . Our first numbers are not only related to the Golombek&#39;s problem, but also computation of the negative order Euler numbers. We compute a few values of the numbers which are given by some tables. We give some applications in Probability and Statistics. That is, special values of mathematical expectation in the binomial distribution and the Bernstein polynomials give us the value of our numbers. Taking derivative of our generating functions, we give partial differential equations and also functional equations. By using these equations, w...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="096acc50d8cea275a4f1fd9b6e08bf75" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":84559077,"asset_id":77100438,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/84559077/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77100438"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77100438"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77100438; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77100438]").text(description); $(".js-view-count[data-work-id=77100438]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77100438; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77100438']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77100438, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "096acc50d8cea275a4f1fd9b6e08bf75" } } $('.js-work-strip[data-work-id=77100438]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77100438,"title":"New families of special numbers for computing negative order Euler numbers","translated_title":"","metadata":{"abstract":"The main purpose of this paper is to construct new families of special numbers with their generating functions. These numbers are related to the many well-known numbers, which are the Bernoulli numbers, the Fibonacci numbers, the Lucas numbers, the Stirling numbers of the second kind and the central factorial numbers. Our other inspiration of this paper is related to the Golombek\u0026#39;s problem golombek“ Aufgabe 1088, El. Math. 49 (1994) 126-127” . Our first numbers are not only related to the Golombek\u0026#39;s problem, but also computation of the negative order Euler numbers. We compute a few values of the numbers which are given by some tables. We give some applications in Probability and Statistics. That is, special values of mathematical expectation in the binomial distribution and the Bernstein polynomials give us the value of our numbers. Taking derivative of our generating functions, we give partial differential equations and also functional equations. By using these equations, w...","publication_date":{"day":19,"month":4,"year":2016,"errors":{}}},"translated_abstract":"The main purpose of this paper is to construct new families of special numbers with their generating functions. These numbers are related to the many well-known numbers, which are the Bernoulli numbers, the Fibonacci numbers, the Lucas numbers, the Stirling numbers of the second kind and the central factorial numbers. Our other inspiration of this paper is related to the Golombek\u0026#39;s problem golombek“ Aufgabe 1088, El. Math. 49 (1994) 126-127” . Our first numbers are not only related to the Golombek\u0026#39;s problem, but also computation of the negative order Euler numbers. We compute a few values of the numbers which are given by some tables. We give some applications in Probability and Statistics. That is, special values of mathematical expectation in the binomial distribution and the Bernstein polynomials give us the value of our numbers. Taking derivative of our generating functions, we give partial differential equations and also functional equations. By using these equations, w...","internal_url":"https://www.academia.edu/77100438/New_families_of_special_numbers_for_computing_negative_order_Euler_numbers","translated_internal_url":"","created_at":"2022-04-20T11:12:47.060-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":146832572,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":84559077,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559077/thumbnails/1.jpg","file_name":"1604.05601v1.pdf","download_url":"https://www.academia.edu/attachments/84559077/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"New_families_of_special_numbers_for_comp.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559077/1604.05601v1-libre.pdf?1650478776=\u0026response-content-disposition=attachment%3B+filename%3DNew_families_of_special_numbers_for_comp.pdf\u0026Expires=1733183071\u0026Signature=DZtW0sJReG9vugwyZzK89r1Vmxv~O0BfwnG5XeMSjgfCLIMLazfCLXNRNag32FlO09gUdJOmPXt6Q4N19f2l0aolZYqhsxpjD-VnX~XR70XiGpFdz4TL~gx-Qnh66OWCZ2naNrEXiJgPI7EPVptdZmJo0gqaLWElyLhL7js1WBR4~dOspSQk7KqATjGH6mD5wMSmwtPuCMPOLeSYycVI47ZXATzaEH3P-s7DAGPbSziQz~c7Twj-RXzPrZ9D0CsZlxAZBnE1MVmiNZRVWQQJ5iYtlB2uG6KvDED8WXnsVP73cSb6242ANaC-e7DYw~ArDlDDJynQlEHODwkBYIInww__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"New_families_of_special_numbers_for_computing_negative_order_Euler_numbers","translated_slug":"","page_count":29,"language":"en","content_type":"Work","owner":{"id":146832572,"first_name":"Yilmaz","middle_initials":null,"last_name":"Simsek","page_name":"YilmazSimsek2","domain_name":"independent","created_at":"2020-02-22T04:58:30.104-08:00","display_name":"Yilmaz Simsek","url":"https://independent.academia.edu/YilmazSimsek2"},"attachments":[{"id":84559077,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559077/thumbnails/1.jpg","file_name":"1604.05601v1.pdf","download_url":"https://www.academia.edu/attachments/84559077/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"New_families_of_special_numbers_for_comp.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559077/1604.05601v1-libre.pdf?1650478776=\u0026response-content-disposition=attachment%3B+filename%3DNew_families_of_special_numbers_for_comp.pdf\u0026Expires=1733183071\u0026Signature=DZtW0sJReG9vugwyZzK89r1Vmxv~O0BfwnG5XeMSjgfCLIMLazfCLXNRNag32FlO09gUdJOmPXt6Q4N19f2l0aolZYqhsxpjD-VnX~XR70XiGpFdz4TL~gx-Qnh66OWCZ2naNrEXiJgPI7EPVptdZmJo0gqaLWElyLhL7js1WBR4~dOspSQk7KqATjGH6mD5wMSmwtPuCMPOLeSYycVI47ZXATzaEH3P-s7DAGPbSziQz~c7Twj-RXzPrZ9D0CsZlxAZBnE1MVmiNZRVWQQJ5iYtlB2uG6KvDED8WXnsVP73cSb6242ANaC-e7DYw~ArDlDDJynQlEHODwkBYIInww__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":84559076,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559076/thumbnails/1.jpg","file_name":"1604.05601v1.pdf","download_url":"https://www.academia.edu/attachments/84559076/download_file","bulk_download_file_name":"New_families_of_special_numbers_for_comp.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559076/1604.05601v1-libre.pdf?1650478773=\u0026response-content-disposition=attachment%3B+filename%3DNew_families_of_special_numbers_for_comp.pdf\u0026Expires=1733183071\u0026Signature=PAT~x-~DUjyIpGr~PC5nYrUdPpQYTLE64CkQZsmCf7XmEccHF~Lz-37m5ReB7IBXzD-I33MvpoLlMJag-mD3klSV3ENcFB7KqeAj3oVbtlnCJu-HiOO~Va3Fpb5KyD6C4zyLOKw09eXZzRX8YJxCzEJzchBhs56RlEQ0t4Yl9x2-i0s94cEWcxwbQ~292AWGAwnNiFIlQvCBoth3BSNQywGGqEik~uTpKuFuRuHwuRnqWEyJ9l~SCpPnVOyDWDlQdVEx2ai1yxdEwzxI0WrhtM1bmAzlEgfvCeFmMnxm-JWzdhrno7PPB~DXR-hHs2fJctcHP35G1PDwIS~eQC4v6Q__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":556845,"name":"Numerical Analysis and Computational Mathematics","url":"https://www.academia.edu/Documents/in/Numerical_Analysis_and_Computational_Mathematics"}],"urls":[{"id":19721506,"url":"https://arxiv.org/pdf/1604.05601v1.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="77100436"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/77100436/Derivation_of_Computational_Formulas_for_certain_class_of_finite_sums_Approach_to_Generating_functions_arising_from_p_adic_integrals_and_special_functions"><img alt="Research paper thumbnail of Derivation of Computational Formulas for certain class of finite sums: Approach to Generating functions arising from p-adic integrals and special functions" class="work-thumbnail" src="https://attachments.academia-assets.com/84559074/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/77100436/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><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The aim of this paper is to construct generating functions for some families of special finite su...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">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...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="4f66cb14d667113a8c6d25619f7822ee" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":84559074,"asset_id":77100436,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/84559074/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77100436"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77100436"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77100436; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77100436]").text(description); $(".js-view-count[data-work-id=77100436]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77100436; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77100436']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77100436, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "4f66cb14d667113a8c6d25619f7822ee" } } $('.js-work-strip[data-work-id=77100436]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77100436,"title":"Derivation of Computational Formulas for certain class of finite sums: Approach to Generating functions arising from p-adic integrals and special functions","translated_title":"","metadata":{"abstract":"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...","publication_date":{"day":22,"month":8,"year":2021,"errors":{}}},"translated_abstract":"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...","internal_url":"https://www.academia.edu/77100436/Derivation_of_Computational_Formulas_for_certain_class_of_finite_sums_Approach_to_Generating_functions_arising_from_p_adic_integrals_and_special_functions","translated_internal_url":"","created_at":"2022-04-20T11:12:46.847-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":146832572,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":84559074,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559074/thumbnails/1.jpg","file_name":"2108.10756v1.pdf","download_url":"https://www.academia.edu/attachments/84559074/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Derivation_of_Computational_Formulas_for.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559074/2108.10756v1-libre.pdf?1650478777=\u0026response-content-disposition=attachment%3B+filename%3DDerivation_of_Computational_Formulas_for.pdf\u0026Expires=1733183071\u0026Signature=Ml1sYJag5F8G3lD7~WJiKIsMRbm1~3mxOGgteDGXk3rGsaI1tZI291~3BzliPjiu4hvaeOjmJuN8jtMvo3dDvlcivOOfFj0fNGf98FVK5MZHSbphd1rAezj92PxpsCQQRRY4nQrbVw72HEVh44kVTXqYO-AGlJnuH~4iF9OYxNlEQji61o9-UMS-m7K4LVR1lB1kv5F8gcUoR4711q1KahiXOFdudAbxcdIfgR0rdudz95j3j8kELWvMg71GA90Jf2vKVX~Ayua-D2W2hTlAkBc8LVP8CarXUYnUtqrd7Oom986zpcfAjLiBF8j7oP1I~jqPf8P8Byd2x-hVrAXJfg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Derivation_of_Computational_Formulas_for_certain_class_of_finite_sums_Approach_to_Generating_functions_arising_from_p_adic_integrals_and_special_functions","translated_slug":"","page_count":33,"language":"en","content_type":"Work","owner":{"id":146832572,"first_name":"Yilmaz","middle_initials":null,"last_name":"Simsek","page_name":"YilmazSimsek2","domain_name":"independent","created_at":"2020-02-22T04:58:30.104-08:00","display_name":"Yilmaz Simsek","url":"https://independent.academia.edu/YilmazSimsek2"},"attachments":[{"id":84559074,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559074/thumbnails/1.jpg","file_name":"2108.10756v1.pdf","download_url":"https://www.academia.edu/attachments/84559074/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Derivation_of_Computational_Formulas_for.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559074/2108.10756v1-libre.pdf?1650478777=\u0026response-content-disposition=attachment%3B+filename%3DDerivation_of_Computational_Formulas_for.pdf\u0026Expires=1733183071\u0026Signature=Ml1sYJag5F8G3lD7~WJiKIsMRbm1~3mxOGgteDGXk3rGsaI1tZI291~3BzliPjiu4hvaeOjmJuN8jtMvo3dDvlcivOOfFj0fNGf98FVK5MZHSbphd1rAezj92PxpsCQQRRY4nQrbVw72HEVh44kVTXqYO-AGlJnuH~4iF9OYxNlEQji61o9-UMS-m7K4LVR1lB1kv5F8gcUoR4711q1KahiXOFdudAbxcdIfgR0rdudz95j3j8kELWvMg71GA90Jf2vKVX~Ayua-D2W2hTlAkBc8LVP8CarXUYnUtqrd7Oom986zpcfAjLiBF8j7oP1I~jqPf8P8Byd2x-hVrAXJfg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":84559075,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559075/thumbnails/1.jpg","file_name":"2108.10756v1.pdf","download_url":"https://www.academia.edu/attachments/84559075/download_file","bulk_download_file_name":"Derivation_of_Computational_Formulas_for.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559075/2108.10756v1-libre.pdf?1650478778=\u0026response-content-disposition=attachment%3B+filename%3DDerivation_of_Computational_Formulas_for.pdf\u0026Expires=1733183071\u0026Signature=E6iUIU6j1XeMFazDDrS-icMY9RMYRJv7yPdo7K6TTnQKy94MGSEJA65qdCS6lJaFTn8XN8CtP4KLGZGOgZi61RrQFs86-vnFAyQSMtK5-MvS62tEexvb-NSCazGaofxj~hTYNs9LANX9kQmmgmdkCGg~RALMmG1ty6W0HKB2xowQdTUBkdvNgTl8oGpj9eJTJcPD4bPeTeNXLVkFEgmPxGz02MhOUYRn-OmEm1Uj72mgWv-iv8slVuuj0PiO3Cnlv20LvJ9ZxnslQ-ZnD-Oi1nNp5bXn2nGtFayCaYDTakY5ArbnQ1BeLamr28UncN0z5iyNPB0yzV1Qo9ODY0Oy9g__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"}],"urls":[{"id":19721505,"url":"https://arxiv.org/pdf/2108.10756v1.pdf"}]}, dispatcherData: dispatcherData }); 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We give p-adic interpretation of this equation.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="ea6d3901a0adc49dd2b98eeaa4648f79" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":84954988,"asset_id":77100435,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/84954988/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77100435"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77100435"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77100435; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77100435]").text(description); $(".js-view-count[data-work-id=77100435]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77100435; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77100435']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77100435, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "ea6d3901a0adc49dd2b98eeaa4648f79" } } $('.js-work-strip[data-work-id=77100435]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77100435,"title":"A p-adic look at the Diophantine equation x^2+11^2k=y^n","translated_title":"","metadata":{"abstract":"We find all solutions of Diophantine equation x^2+11^2k = y^n where x\u0026gt;=1, y\u0026gt;=1, n\u0026gt;=3 and k is natural number. We give p-adic interpretation of this equation.","publication_date":{"day":27,"month":12,"year":2011,"errors":{}}},"translated_abstract":"We find all solutions of Diophantine equation x^2+11^2k = y^n where x\u0026gt;=1, y\u0026gt;=1, n\u0026gt;=3 and k is natural number. We give p-adic interpretation of this equation.","internal_url":"https://www.academia.edu/77100435/A_p_adic_look_at_the_Diophantine_equation_x_2_11_2k_y_n","translated_internal_url":"","created_at":"2022-04-20T11:12:46.602-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":146832572,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":84954988,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84954988/thumbnails/1.jpg","file_name":"1112.pdf","download_url":"https://www.academia.edu/attachments/84954988/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"A_p_adic_look_at_the_Diophantine_equatio.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84954988/1112-libre.pdf?1650959490=\u0026response-content-disposition=attachment%3B+filename%3DA_p_adic_look_at_the_Diophantine_equatio.pdf\u0026Expires=1733183071\u0026Signature=YJ4KkVT6pM5jVLxEZJLk3dndmoGXR0daUObakYwU9eZrFvW9tmTwU37c6ftSMgadTjSCJC0mWwZg8TvtgMmu7Anafy~YLJ2X5jFFZVRVTaI1cTZTJ75JpRIWVedjg8sNfgMBYaWOpA2SiShayk-Z~cqh83zQrduFLOlfqAC7JLQqYp~DqrWJK42nGDINLT~JMxXnqFTrCVKwug7YfkdnV~PLryofjw0~kYG~fzlIBkdgrPgkFVTBRvv3toOAqyPewo4kKafVa8fJB67IK4hztOINOHy9w3ZLlsh~1ocFLjAfx7UQ8ElVKwR6ZAJ2S2AO7RggmRRPT73Pf2vruYKz6Q__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"A_p_adic_look_at_the_Diophantine_equation_x_2_11_2k_y_n","translated_slug":"","page_count":4,"language":"en","content_type":"Work","owner":{"id":146832572,"first_name":"Yilmaz","middle_initials":null,"last_name":"Simsek","page_name":"YilmazSimsek2","domain_name":"independent","created_at":"2020-02-22T04:58:30.104-08:00","display_name":"Yilmaz Simsek","url":"https://independent.academia.edu/YilmazSimsek2"},"attachments":[{"id":84954988,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84954988/thumbnails/1.jpg","file_name":"1112.pdf","download_url":"https://www.academia.edu/attachments/84954988/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"A_p_adic_look_at_the_Diophantine_equatio.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84954988/1112-libre.pdf?1650959490=\u0026response-content-disposition=attachment%3B+filename%3DA_p_adic_look_at_the_Diophantine_equatio.pdf\u0026Expires=1733183071\u0026Signature=YJ4KkVT6pM5jVLxEZJLk3dndmoGXR0daUObakYwU9eZrFvW9tmTwU37c6ftSMgadTjSCJC0mWwZg8TvtgMmu7Anafy~YLJ2X5jFFZVRVTaI1cTZTJ75JpRIWVedjg8sNfgMBYaWOpA2SiShayk-Z~cqh83zQrduFLOlfqAC7JLQqYp~DqrWJK42nGDINLT~JMxXnqFTrCVKwug7YfkdnV~PLryofjw0~kYG~fzlIBkdgrPgkFVTBRvv3toOAqyPewo4kKafVa8fJB67IK4hztOINOHy9w3ZLlsh~1ocFLjAfx7UQ8ElVKwR6ZAJ2S2AO7RggmRRPT73Pf2vruYKz6Q__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":194130,"name":"PARTIAL DIFFERENTIAL EQUATION","url":"https://www.academia.edu/Documents/in/PARTIAL_DIFFERENTIAL_EQUATION"},{"id":506482,"name":"Function approximation","url":"https://www.academia.edu/Documents/in/Function_approximation"},{"id":939537,"name":"Boundary Value Problem","url":"https://www.academia.edu/Documents/in/Boundary_Value_Problem"},{"id":3641544,"name":"Diophantine Equation","url":"https://www.academia.edu/Documents/in/Diophantine_Equation"}],"urls":[{"id":19721504,"url":"https://core.ac.uk/download/pdf/2244862.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="77100434"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/77100434/Identities_and_relations_related_to_the_numbers_of_special_words_derived_from_special_series_with_Dirichlet_convolution"><img alt="Research paper thumbnail of Identities and relations related to the numbers of special words derived from special series with Dirichlet convolution" class="work-thumbnail" src="https://attachments.academia-assets.com/84559072/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/77100434/Identities_and_relations_related_to_the_numbers_of_special_words_derived_from_special_series_with_Dirichlet_convolution">Identities and relations related to the numbers of special words derived from special series with Dirichlet convolution</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The aim of this paper is to define some new number-theoretic functions including necklaces polyno...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The aim of this paper is to define some new number-theoretic functions including necklaces polynomials and the numbers of special words such as Lyndon words. By using Dirichlet convolution formula with well-known number-theoretic functions, we derive some new identities and relations associated with Dirichlet series, Lambert series, and also the family of zeta functions including the Riemann zeta functions and polylogarithm functions. By using analytic (meromorphic) continuation of zeta functions, we also derive identities and formulas including Bernoulli numbers and Apostol-Bernoulli numbers. Moreover, we give relations between number-theoretic functions and the Fourier expansion of the Eisenstein series. Finally, we give some observations and remarks on these functions.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="418c97e8a16bc65033893e247f271285" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":84559072,"asset_id":77100434,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/84559072/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77100434"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77100434"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77100434; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77100434]").text(description); $(".js-view-count[data-work-id=77100434]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77100434; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77100434']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77100434, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "418c97e8a16bc65033893e247f271285" } } $('.js-work-strip[data-work-id=77100434]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77100434,"title":"Identities and relations related to the numbers of special words derived from special series with Dirichlet convolution","translated_title":"","metadata":{"abstract":"The aim of this paper is to define some new number-theoretic functions including necklaces polynomials and the numbers of special words such as Lyndon words. By using Dirichlet convolution formula with well-known number-theoretic functions, we derive some new identities and relations associated with Dirichlet series, Lambert series, and also the family of zeta functions including the Riemann zeta functions and polylogarithm functions. By using analytic (meromorphic) continuation of zeta functions, we also derive identities and formulas including Bernoulli numbers and Apostol-Bernoulli numbers. Moreover, we give relations between number-theoretic functions and the Fourier expansion of the Eisenstein series. Finally, we give some observations and remarks on these functions.","publication_date":{"day":25,"month":6,"year":2018,"errors":{}}},"translated_abstract":"The aim of this paper is to define some new number-theoretic functions including necklaces polynomials and the numbers of special words such as Lyndon words. By using Dirichlet convolution formula with well-known number-theoretic functions, we derive some new identities and relations associated with Dirichlet series, Lambert series, and also the family of zeta functions including the Riemann zeta functions and polylogarithm functions. By using analytic (meromorphic) continuation of zeta functions, we also derive identities and formulas including Bernoulli numbers and Apostol-Bernoulli numbers. Moreover, we give relations between number-theoretic functions and the Fourier expansion of the Eisenstein series. Finally, we give some observations and remarks on these functions.","internal_url":"https://www.academia.edu/77100434/Identities_and_relations_related_to_the_numbers_of_special_words_derived_from_special_series_with_Dirichlet_convolution","translated_internal_url":"","created_at":"2022-04-20T11:12:46.368-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":146832572,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":84559072,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559072/thumbnails/1.jpg","file_name":"1806.09480v1.pdf","download_url":"https://www.academia.edu/attachments/84559072/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Identities_and_relations_related_to_the.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559072/1806.09480v1-libre.pdf?1650478776=\u0026response-content-disposition=attachment%3B+filename%3DIdentities_and_relations_related_to_the.pdf\u0026Expires=1733183071\u0026Signature=c9tyxvNusjRf8VW0IaYG4p-zDXqO-X7zed5LP3MPy7NlDRU9aPKIYE6tske6AW21lwPpVd4EhDcEgLjZQh1Q~0R6hy~akGX7Argjblqzw4wMr2rlVJSx2Dpyx0-jRyvsjRLZCJI~gRqZVxpLl6eCXiTGaCPW5NS3GRMpmSpZ1Zss0yA~DL~sewfLaURlq11Quz5dPuqVac8qXiE0qnuEZ5HZ7Abd1Ue3YLicbOlpC8wZb2JfA2NMotaCU8Adw9Lbl0HR4QslDsyDWmE7tx5Ff5flSZ7qJIIy7A7fz-XrwQaXE4jUwg4JTs17yWLM0kNdemt9751hn-hYN2-bZgbZkw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Identities_and_relations_related_to_the_numbers_of_special_words_derived_from_special_series_with_Dirichlet_convolution","translated_slug":"","page_count":20,"language":"en","content_type":"Work","owner":{"id":146832572,"first_name":"Yilmaz","middle_initials":null,"last_name":"Simsek","page_name":"YilmazSimsek2","domain_name":"independent","created_at":"2020-02-22T04:58:30.104-08:00","display_name":"Yilmaz Simsek","url":"https://independent.academia.edu/YilmazSimsek2"},"attachments":[{"id":84559072,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559072/thumbnails/1.jpg","file_name":"1806.09480v1.pdf","download_url":"https://www.academia.edu/attachments/84559072/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Identities_and_relations_related_to_the.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559072/1806.09480v1-libre.pdf?1650478776=\u0026response-content-disposition=attachment%3B+filename%3DIdentities_and_relations_related_to_the.pdf\u0026Expires=1733183071\u0026Signature=c9tyxvNusjRf8VW0IaYG4p-zDXqO-X7zed5LP3MPy7NlDRU9aPKIYE6tske6AW21lwPpVd4EhDcEgLjZQh1Q~0R6hy~akGX7Argjblqzw4wMr2rlVJSx2Dpyx0-jRyvsjRLZCJI~gRqZVxpLl6eCXiTGaCPW5NS3GRMpmSpZ1Zss0yA~DL~sewfLaURlq11Quz5dPuqVac8qXiE0qnuEZ5HZ7Abd1Ue3YLicbOlpC8wZb2JfA2NMotaCU8Adw9Lbl0HR4QslDsyDWmE7tx5Ff5flSZ7qJIIy7A7fz-XrwQaXE4jUwg4JTs17yWLM0kNdemt9751hn-hYN2-bZgbZkw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":84559073,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559073/thumbnails/1.jpg","file_name":"1806.09480v1.pdf","download_url":"https://www.academia.edu/attachments/84559073/download_file","bulk_download_file_name":"Identities_and_relations_related_to_the.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559073/1806.09480v1-libre.pdf?1650478777=\u0026response-content-disposition=attachment%3B+filename%3DIdentities_and_relations_related_to_the.pdf\u0026Expires=1733183071\u0026Signature=XjEz6AsJkkmCuNcGgw3wu0Ru5FLNmORnbFkVwi5k9uSBf2~dpRMugCF-Q23zRgbUfVD1NlxRL2LvLHiXZtGM-KkfjPdYRzitaYaytqZzqmk8BQc-QR5WdGSBbXXPuFCQ29Ah7MPZDh2QfZXxgZi-u3SzXVGRgl5OUhWm6KtJMrqaF-wd50JCQNoVsEHvd3NdCInPffxNt~ov9WF4j~nPa8FVEquJ0ixZtGRyk-ILfabNaEW~ICNdbL~FtzQzKwZZnBuwJDtFB4Wopu-jdTOEC013HcTHUzyl8GEepDLhLIjxVxB8lErbqJlivcsGwkjNXe3RIPO0vrC-1ZJwMWfYVw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"}],"urls":[{"id":19721503,"url":"https://arxiv.org/pdf/1806.09480v1.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="77100433"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/77100433/Combinatorial_identities_associated_with_new_families_of_the_numbers_and_polynomials_and_their_approximation_values"><img alt="Research paper thumbnail of Combinatorial identities associated with new families of the numbers and polynomials and their approximation values" class="work-thumbnail" src="https://attachments.academia-assets.com/84559070/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/77100433/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><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Recently, the numbers Y_n(λ ) and the polynomials Y_n(x,λ) have been introduced by the second aut...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Recently, the numbers Y_n(λ ) and the polynomials Y_n(x,λ) have been introduced by the second author [22]. The purpose of this paper is to construct higher-order of these numbers and polynomials with their generating functions. By using these generating functions with their functional equations and derivative equations, we derive various identities and relations including two recurrence relations, Vandermonde type convolution formula, combinatorial sums, the Bernstein basis functions, and also some well known families of special numbers and their interpolation functions such as the Apostol--Bernoulli numbers, the Apostol--Euler numbers, the Stirling numbers of the first kind, and the zeta type function. Finally, by using Stirling&#39;s approximation for factorials, we investigate some approximation values of the special case of the numbers Y_n( λ) .</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="c8001a5ca7c18c80fc7764cc75e966b5" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":84559070,"asset_id":77100433,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/84559070/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77100433"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77100433"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77100433; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77100433]").text(description); $(".js-view-count[data-work-id=77100433]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77100433; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77100433']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77100433, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "c8001a5ca7c18c80fc7764cc75e966b5" } } $('.js-work-strip[data-work-id=77100433]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77100433,"title":"Combinatorial identities associated with new families of the numbers and polynomials and their approximation values","translated_title":"","metadata":{"abstract":"Recently, the numbers Y_n(λ ) and the polynomials Y_n(x,λ) have been introduced by the second author [22]. The purpose of this paper is to construct higher-order of these numbers and polynomials with their generating functions. By using these generating functions with their functional equations and derivative equations, we derive various identities and relations including two recurrence relations, Vandermonde type convolution formula, combinatorial sums, the Bernstein basis functions, and also some well known families of special numbers and their interpolation functions such as the Apostol--Bernoulli numbers, the Apostol--Euler numbers, the Stirling numbers of the first kind, and the zeta type function. Finally, by using Stirling\u0026#39;s approximation for factorials, we investigate some approximation values of the special case of the numbers Y_n( λ) .","publication_date":{"day":30,"month":10,"year":2017,"errors":{}}},"translated_abstract":"Recently, the numbers Y_n(λ ) and the polynomials Y_n(x,λ) have been introduced by the second author [22]. The purpose of this paper is to construct higher-order of these numbers and polynomials with their generating functions. By using these generating functions with their functional equations and derivative equations, we derive various identities and relations including two recurrence relations, Vandermonde type convolution formula, combinatorial sums, the Bernstein basis functions, and also some well known families of special numbers and their interpolation functions such as the Apostol--Bernoulli numbers, the Apostol--Euler numbers, the Stirling numbers of the first kind, and the zeta type function. Finally, by using Stirling\u0026#39;s approximation for factorials, we investigate some approximation values of the special case of the numbers Y_n( λ) .","internal_url":"https://www.academia.edu/77100433/Combinatorial_identities_associated_with_new_families_of_the_numbers_and_polynomials_and_their_approximation_values","translated_internal_url":"","created_at":"2022-04-20T11:12:46.097-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":146832572,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":84559070,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559070/thumbnails/1.jpg","file_name":"1711.00850v1.pdf","download_url":"https://www.academia.edu/attachments/84559070/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Combinatorial_identities_associated_with.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559070/1711.00850v1-libre.pdf?1650478773=\u0026response-content-disposition=attachment%3B+filename%3DCombinatorial_identities_associated_with.pdf\u0026Expires=1733183071\u0026Signature=ONDBK8G7u4ak1rSfk12hys8lPG3uVcVvH7tzO3brjKUoO4KbDpCQm~~euLmGYYmGPBFs4Zxs1r6PMkcrwyJNzd4CpjyA5tGsAko8QTqHHjpGa4QuLCt9UO7FCSNgnz~WNzgngIqYwTQ71sggP0E9ZzDCnPHceh5STALc1ABD8oxno3yCHuX8DWamXtkHvsgFDZaTPxegfVmixwnBlvQzhgkpHw-1WSIDEuD0B-F8m6NULZDhq8O9rMZWMKuRctA2M6awOzXH-5yVvwIZ-JIexIDMsr4eW4F~4VtJzpaSE3m-bdhQXzGBrQS8ye48uyrTgUVFfve5JQSBffqMPzhswA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Combinatorial_identities_associated_with_new_families_of_the_numbers_and_polynomials_and_their_approximation_values","translated_slug":"","page_count":17,"language":"en","content_type":"Work","owner":{"id":146832572,"first_name":"Yilmaz","middle_initials":null,"last_name":"Simsek","page_name":"YilmazSimsek2","domain_name":"independent","created_at":"2020-02-22T04:58:30.104-08:00","display_name":"Yilmaz Simsek","url":"https://independent.academia.edu/YilmazSimsek2"},"attachments":[{"id":84559070,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559070/thumbnails/1.jpg","file_name":"1711.00850v1.pdf","download_url":"https://www.academia.edu/attachments/84559070/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Combinatorial_identities_associated_with.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559070/1711.00850v1-libre.pdf?1650478773=\u0026response-content-disposition=attachment%3B+filename%3DCombinatorial_identities_associated_with.pdf\u0026Expires=1733183071\u0026Signature=ONDBK8G7u4ak1rSfk12hys8lPG3uVcVvH7tzO3brjKUoO4KbDpCQm~~euLmGYYmGPBFs4Zxs1r6PMkcrwyJNzd4CpjyA5tGsAko8QTqHHjpGa4QuLCt9UO7FCSNgnz~WNzgngIqYwTQ71sggP0E9ZzDCnPHceh5STALc1ABD8oxno3yCHuX8DWamXtkHvsgFDZaTPxegfVmixwnBlvQzhgkpHw-1WSIDEuD0B-F8m6NULZDhq8O9rMZWMKuRctA2M6awOzXH-5yVvwIZ-JIexIDMsr4eW4F~4VtJzpaSE3m-bdhQXzGBrQS8ye48uyrTgUVFfve5JQSBffqMPzhswA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":84559071,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559071/thumbnails/1.jpg","file_name":"1711.00850v1.pdf","download_url":"https://www.academia.edu/attachments/84559071/download_file","bulk_download_file_name":"Combinatorial_identities_associated_with.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559071/1711.00850v1-libre.pdf?1650478774=\u0026response-content-disposition=attachment%3B+filename%3DCombinatorial_identities_associated_with.pdf\u0026Expires=1733183071\u0026Signature=Gu3ZEQE9EY4gYajhcvGjSijtkB1r--iMF0Aca7EDyvnxeK0RhG6J3pioIFyE45EOMFgl~mVfa3NPND8RUmBmpQT8n7XIUlEDMg5KzogdhYaJAnGIpgRa01FUgIFGamQVHtGpNMump4d3kC732YnkmlTZEejrGgSE-81q5ehYaBByPB6MONRFKaRxQWtO9PWC6JyFKb76McxZGnCFtTe9h8Bz5jzucu5mKAs5WBuCLfR0V9h261UTvikm4D6SFmvT-S78QyFGS9WgEbMTwfWaYYEZvoyAmNDqoFELpZGp0oM4AnbyvVMoiJ9AageVu4KqTbDlP5C2OkP0TJ-R4W-3jQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"}],"urls":[{"id":19721502,"url":"https://arxiv.org/pdf/1711.00850v1.pdf"}]}, dispatcherData: dispatcherData }); 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Finally, we will give some relations between these numbers anf polynomials</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="3c8932b153413c648e327a25c1aaf005" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":84559068,"asset_id":77100432,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/84559068/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77100432"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77100432"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77100432; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77100432]").text(description); $(".js-view-count[data-work-id=77100432]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77100432; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77100432']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77100432, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "3c8932b153413c648e327a25c1aaf005" } } $('.js-work-strip[data-work-id=77100432]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77100432,"title":"Complete sums of products of (h,q)-extension of Euler numbers and polynomials","translated_title":"","metadata":{"abstract":"In this paper we investigate some interesting of the (h,q)-extension of Euler numbers and polynomials. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="77100431"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/77100431/Generating_functions_for_finite_sums_involving_higher_powers_of_binomial_coefficients_Analysis_of_hypergeometric_functions_including_new_families_of_polynomials_and_numbers"><img alt="Research paper thumbnail of Generating functions for finite sums involving higher powers of binomial coefficients: Analysis of hypergeometric functions including new families of polynomials and numbers" class="work-thumbnail" src="https://attachments.academia-assets.com/84559066/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/77100431/Generating_functions_for_finite_sums_involving_higher_powers_of_binomial_coefficients_Analysis_of_hypergeometric_functions_including_new_families_of_polynomials_and_numbers">Generating functions for finite sums involving higher powers of binomial coefficients: Analysis of hypergeometric functions including new families of polynomials and numbers</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The origin of this study is based on not only explicit formulas of finite sums involving higher p...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The origin of this study is based on not only explicit formulas of finite sums involving higher powers of binomial coefficients, but also explicit evaluations of generating functions for this sums. It should be emphasized that this study contains both new results and literature surveys about some of the related results that have existed so far. With the aid of hypergeometric function, generating functions for a new family of the combinatorial numbers, related to finite sums of powers of binomial coefficients, are constructed. By using these generating functions, a number of new identities have been obtained and at the same time previously well-known formulas and identities have been generalized. Moreover, on this occasion, we identify new families of polynomials including some families of well-known numbers such as Bernoulli numbers, Euler numbers, Stirling numbers, Franel numbers, Catalan numbers, Changhee numbers, Daehee numbers and the others, and also for the polynomials such as...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="3dd2e18c7781a95c56c085886d2269b3" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":84559066,"asset_id":77100431,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/84559066/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77100431"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77100431"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77100431; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77100431]").text(description); $(".js-view-count[data-work-id=77100431]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77100431; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77100431']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77100431, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "3dd2e18c7781a95c56c085886d2269b3" } } $('.js-work-strip[data-work-id=77100431]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77100431,"title":"Generating functions for finite sums involving higher powers of binomial coefficients: Analysis of hypergeometric functions including new families of polynomials and numbers","translated_title":"","metadata":{"abstract":"The origin of this study is based on not only explicit formulas of finite sums involving higher powers of binomial coefficients, but also explicit evaluations of generating functions for this sums. 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Moreover, on this occasion, we identify new families of polynomials including some families of well-known numbers such as Bernoulli numbers, Euler numbers, Stirling numbers, Franel numbers, Catalan numbers, Changhee numbers, Daehee numbers and the others, and also for the polynomials such as...","publication_date":{"day":4,"month":1,"year":2019,"errors":{}}},"translated_abstract":"The origin of this study is based on not only explicit formulas of finite sums involving higher powers of binomial coefficients, but also explicit evaluations of generating functions for this sums. It should be emphasized that this study contains both new results and literature surveys about some of the related results that have existed so far. With the aid of hypergeometric function, generating functions for a new family of the combinatorial numbers, related to finite sums of powers of binomial coefficients, are constructed. By using these generating functions, a number of new identities have been obtained and at the same time previously well-known formulas and identities have been generalized. Moreover, on this occasion, we identify new families of polynomials including some families of well-known numbers such as Bernoulli numbers, Euler numbers, Stirling numbers, Franel numbers, Catalan numbers, Changhee numbers, Daehee numbers and the others, and also for the polynomials such as...","internal_url":"https://www.academia.edu/77100431/Generating_functions_for_finite_sums_involving_higher_powers_of_binomial_coefficients_Analysis_of_hypergeometric_functions_including_new_families_of_polynomials_and_numbers","translated_internal_url":"","created_at":"2022-04-20T11:12:45.594-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":146832572,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":84559066,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559066/thumbnails/1.jpg","file_name":"1901.02912v1.pdf","download_url":"https://www.academia.edu/attachments/84559066/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Generating_functions_for_finite_sums_inv.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559066/1901.02912v1-libre.pdf?1650478775=\u0026response-content-disposition=attachment%3B+filename%3DGenerating_functions_for_finite_sums_inv.pdf\u0026Expires=1733183071\u0026Signature=EdD7~Bsh9xsvqNam5c3JPtq4JFPCpHHaPZtupA86JooqoJHrf7fahUHlEjqjnsNEl6PeMtT7roJ0MzPdvtGSCk04BYXy1YpkhlM3uT4yoNvIj~ePSIjdkKall~4WmUw9-op7IboY3xOs5EaN26LzhVuyTWD-McVib2ecrqherV0IMCKw491h6FPVTXyDcqWFIC~gKaUDBeC5x7noAM85s5sNRpSQbsEFLJrQS94mn0TODpEGp~FR2NR4qXBZaX5gaRj~usSEoGbyWbJZAm3iRTvAb-rDjHGKz~gi3U1go5wyGkId9VofzMJUFuDVubCu5GF~Xwj3nOb8E544BmREEA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Generating_functions_for_finite_sums_involving_higher_powers_of_binomial_coefficients_Analysis_of_hypergeometric_functions_including_new_families_of_polynomials_and_numbers","translated_slug":"","page_count":36,"language":"en","content_type":"Work","owner":{"id":146832572,"first_name":"Yilmaz","middle_initials":null,"last_name":"Simsek","page_name":"YilmazSimsek2","domain_name":"independent","created_at":"2020-02-22T04:58:30.104-08:00","display_name":"Yilmaz Simsek","url":"https://independent.academia.edu/YilmazSimsek2"},"attachments":[{"id":84559066,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559066/thumbnails/1.jpg","file_name":"1901.02912v1.pdf","download_url":"https://www.academia.edu/attachments/84559066/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Generating_functions_for_finite_sums_inv.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559066/1901.02912v1-libre.pdf?1650478775=\u0026response-content-disposition=attachment%3B+filename%3DGenerating_functions_for_finite_sums_inv.pdf\u0026Expires=1733183071\u0026Signature=EdD7~Bsh9xsvqNam5c3JPtq4JFPCpHHaPZtupA86JooqoJHrf7fahUHlEjqjnsNEl6PeMtT7roJ0MzPdvtGSCk04BYXy1YpkhlM3uT4yoNvIj~ePSIjdkKall~4WmUw9-op7IboY3xOs5EaN26LzhVuyTWD-McVib2ecrqherV0IMCKw491h6FPVTXyDcqWFIC~gKaUDBeC5x7noAM85s5sNRpSQbsEFLJrQS94mn0TODpEGp~FR2NR4qXBZaX5gaRj~usSEoGbyWbJZAm3iRTvAb-rDjHGKz~gi3U1go5wyGkId9VofzMJUFuDVubCu5GF~Xwj3nOb8E544BmREEA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":84559067,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559067/thumbnails/1.jpg","file_name":"1901.02912v1.pdf","download_url":"https://www.academia.edu/attachments/84559067/download_file","bulk_download_file_name":"Generating_functions_for_finite_sums_inv.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559067/1901.02912v1-libre.pdf?1650478776=\u0026response-content-disposition=attachment%3B+filename%3DGenerating_functions_for_finite_sums_inv.pdf\u0026Expires=1733183071\u0026Signature=FuIiZx9lRprinbgLbSWFubNlkq7kzJzxxRz2TBjgU7tkJj7tPD1hIOy-BLjFL5gAUu5HaNXwmzynxfBNqnxSfD2RwQN1thoSmvvLuirFlbDphu~kbK9rUOCmlcfeDfgMm-4s1q3oG2ix~BAauGCUM8eo3Dr15Cxs0yM7ykOhEtPJRytw7LSkdHG2fJmk4SdTAy8kL1JXieKKX6Kcnzm-TInhFuiaFcKsdnr43FenQnACN13DcAZpUhQaVIpnOF5jU5tY7uVI-RpaQsRs4ilusmnJYP8BRiObkcXTqb7QqWRdZOZXZsSdsm0dVVI2bdnsZcb0iCyKCSnfZ9O5xWEKDA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":305,"name":"Applied Mathematics","url":"https://www.academia.edu/Documents/in/Applied_Mathematics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":518958,"name":"Mathematical Analysis and Applications","url":"https://www.academia.edu/Documents/in/Mathematical_Analysis_and_Applications"},{"id":1131312,"name":"Academic","url":"https://www.academia.edu/Documents/in/Academic"},{"id":1237788,"name":"Electrical And Electronic Engineering","url":"https://www.academia.edu/Documents/in/Electrical_And_Electronic_Engineering"}],"urls":[{"id":19721500,"url":"https://arxiv.org/pdf/1901.02912v1.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="77100430"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/77100430/Functional_equations_from_generating_functions_a_novel_approach_to_deriving_identities_for_the_Bernstein_basis_functions"><img alt="Research paper thumbnail of Functional equations from generating functions: a novel approach to deriving identities for the Bernstein basis functions" class="work-thumbnail" src="https://attachments.academia-assets.com/84559064/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/77100430/Functional_equations_from_generating_functions_a_novel_approach_to_deriving_identities_for_the_Bernstein_basis_functions">Functional equations from generating functions: a novel approach to deriving identities for the Bernstein basis functions</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The main aim of this paper is to provide a novel approach to deriving identities for the Bernstei...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The main aim of this paper is to provide a novel approach to deriving identities for the Bernstein polynomials using functional equations. We derive various functional equations and differential equations using generating functions. Applying these equations, we give new proofs for some standard identities for the Bernstein basis functions, including formulas for sums, alternating sums, recursion, subdivision, degree raising, differentiation and a formula for the monomials in terms of the Bernstein basis functions. We also derive many new identities for the Bernstein basis functions based on this approach. Moreover, by applying the Laplace transform to the generating functions for the Bernstein basis functions, we obtain some interesting series representations for the Bernstein basis functions.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="17b59aa3102458f819acb536909f66b0" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":84559064,"asset_id":77100430,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/84559064/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77100430"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77100430"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77100430; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77100430]").text(description); $(".js-view-count[data-work-id=77100430]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77100430; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77100430']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77100430, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "17b59aa3102458f819acb536909f66b0" } } $('.js-work-strip[data-work-id=77100430]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77100430,"title":"Functional equations from generating functions: a novel approach to deriving identities for the Bernstein basis functions","translated_title":"","metadata":{"abstract":"The main aim of this paper is to provide a novel approach to deriving identities for the Bernstein polynomials using functional equations. We derive various functional equations and differential equations using generating functions. Applying these equations, we give new proofs for some standard identities for the Bernstein basis functions, including formulas for sums, alternating sums, recursion, subdivision, degree raising, differentiation and a formula for the monomials in terms of the Bernstein basis functions. We also derive many new identities for the Bernstein basis functions based on this approach. Moreover, by applying the Laplace transform to the generating functions for the Bernstein basis functions, we obtain some interesting series representations for the Bernstein basis functions.","publication_date":{"day":21,"month":11,"year":2011,"errors":{}}},"translated_abstract":"The main aim of this paper is to provide a novel approach to deriving identities for the Bernstein polynomials using functional equations. We derive various functional equations and differential equations using generating functions. 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data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/101102956/Derivation_of_computational_formulas_for_Changhee_polynomials_and_their_functional_and_differential_equations"><img alt="Research paper thumbnail of Derivation of computational formulas for Changhee polynomials and their functional and differential equations" class="work-thumbnail" src="https://attachments.academia-assets.com/101735105/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/101102956/Derivation_of_computational_formulas_for_Changhee_polynomials_and_their_functional_and_differential_equations">Derivation of computational formulas for Changhee polynomials and their functional and differential equations</a></div><div class="wp-workCard_item"><span>Journal of Inequalities and Applications</span><span>, 2020</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The goal of this paper is to demonstrate many explicit computational formulas and relations invol...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The goal of this paper is to demonstrate many explicit computational formulas and relations involving the Changhee polynomials and numbers and their differential equations with the help of functional equations and partial derivative equations for generating functions of these polynomials and numbers. These formulas also include the Euler polynomials, the Stirling numbers, the Bernoulli numbers and polynomials of the second kind, the Changhee polynomials of higher order, and the Daehee polynomials of higher order, which are among the well known polynomial families. By using PDEs of these generating functions, not only some recurrence relations for derivative formulas of the Changhee polynomials of higher order, but also two open problems for partial derivative equations for generating functions are given. Moreover, by using functional equations of the generating functions, two inequalities including combinatorial sums, the Changhee numbers of negative order, and the Stirling numbers ...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="fb5c91cfb2317c190fa2403d0354436e" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":101735105,"asset_id":101102956,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/101735105/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="101102956"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="101102956"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 101102956; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=101102956]").text(description); $(".js-view-count[data-work-id=101102956]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 101102956; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='101102956']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 101102956, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "fb5c91cfb2317c190fa2403d0354436e" } } $('.js-work-strip[data-work-id=101102956]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":101102956,"title":"Derivation of computational formulas for Changhee polynomials and their functional and differential equations","translated_title":"","metadata":{"abstract":"The goal of this paper is to demonstrate many explicit computational formulas and relations involving the Changhee polynomials and numbers and their differential equations with the help of functional equations and partial derivative equations for generating functions of these polynomials and numbers. 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Moreover, by using functional equations of the generating functions, two inequalities including combinatorial sums, the Changhee numbers of negative order, and the Stirling numbers ...","publisher":"Springer Science and Business Media LLC","publication_date":{"day":null,"month":null,"year":2020,"errors":{}},"publication_name":"Journal of Inequalities and Applications"},"translated_abstract":"The goal of this paper is to demonstrate many explicit computational formulas and relations involving the Changhee polynomials and numbers and their differential equations with the help of functional equations and partial derivative equations for generating functions of these polynomials and numbers. These formulas also include the Euler polynomials, the Stirling numbers, the Bernoulli numbers and polynomials of the second kind, the Changhee polynomials of higher order, and the Daehee polynomials of higher order, which are among the well known polynomial families. By using PDEs of these generating functions, not only some recurrence relations for derivative formulas of the Changhee polynomials of higher order, but also two open problems for partial derivative equations for generating functions are given. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="96265213"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/96265213/The_actions_on_the_generating_functions_for_the_family_of_the_generalized_Bernoulli_polynomials"><img alt="Research paper thumbnail of The actions on the generating functions for the family of the generalized Bernoulli polynomials" class="work-thumbnail" src="https://attachments.academia-assets.com/98212637/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/96265213/The_actions_on_the_generating_functions_for_the_family_of_the_generalized_Bernoulli_polynomials">The actions on the generating functions for the family of the generalized Bernoulli polynomials</a></div><div class="wp-workCard_item"><span>Filomat</span><span>, 2017</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In this paper, we study the generalization Bernoulli numbers and polynomials attached to a period...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">In this paper, we study the generalization Bernoulli numbers and polynomials attached to a periodic group homomorphism from a finite cyclic group to the set of complex numbers and derive new periodic group homomorphism by using a fixed periodic group homomorphism. Hence, we obtain not only multiplication formulas, but also some new identities for the generalized Bernoulli polynomials attached to a periodic group homomorphism.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="5f682bc9d1932d836254a49204217f7e" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":98212637,"asset_id":96265213,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/98212637/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="96265213"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="96265213"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 96265213; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=96265213]").text(description); $(".js-view-count[data-work-id=96265213]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 96265213; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='96265213']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 96265213, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "5f682bc9d1932d836254a49204217f7e" } } $('.js-work-strip[data-work-id=96265213]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":96265213,"title":"The actions on the generating functions for the family of the generalized Bernoulli polynomials","translated_title":"","metadata":{"abstract":"In this paper, we study the generalization Bernoulli numbers and polynomials attached to a periodic group homomorphism from a finite cyclic group to the set of complex numbers and derive new periodic group homomorphism by using a fixed periodic group homomorphism. 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Hence, we obtain not only multiplication formulas, but also some new identities for the generalized Bernoulli polynomials attached to a periodic group homomorphism.","internal_url":"https://www.academia.edu/96265213/The_actions_on_the_generating_functions_for_the_family_of_the_generalized_Bernoulli_polynomials","translated_internal_url":"","created_at":"2023-02-03T22:29:11.171-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":146832572,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":98212637,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/98212637/thumbnails/1.jpg","file_name":"61c541c2a67ae67c82ee1f2217032b43102b.pdf","download_url":"https://www.academia.edu/attachments/98212637/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"The_actions_on_the_generating_functions.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/98212637/61c541c2a67ae67c82ee1f2217032b43102b-libre.pdf?1675494178=\u0026response-content-disposition=attachment%3B+filename%3DThe_actions_on_the_generating_functions.pdf\u0026Expires=1733183070\u0026Signature=SD0OcLyf2Z~hCfkUygUcOQrcEwuX5SiBiRL9CID0efX7NyuCclIwcI3szTnKteWbUJf-OEjw51FWDH9FG1F81WBVvd45Fa2xEnAfwt0E~qIcAHPUkDPi01OrEvXnrFPDI9Nn2DxSBDPrkqhW1xkoUVhQLpheHBARIHzOfli8a1Fj2imWTG01czmkpfSdU~5NyR00f-84oZJEtQz4Qbc4grli56KAn-Oi71DQNeUTQMjJ8EmpG9YmXrDPhaCoeUztEebdGwQa7RVHh1L47F3kYmwd0S5wGVeGA3y~WXLVGjcznItlG~knEUcx4Q2O~DaMHzA74Oq7cvJv2a272TRrGQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"The_actions_on_the_generating_functions_for_the_family_of_the_generalized_Bernoulli_polynomials","translated_slug":"","page_count":10,"language":"en","content_type":"Work","owner":{"id":146832572,"first_name":"Yilmaz","middle_initials":null,"last_name":"Simsek","page_name":"YilmazSimsek2","domain_name":"independent","created_at":"2020-02-22T04:58:30.104-08:00","display_name":"Yilmaz Simsek","url":"https://independent.academia.edu/YilmazSimsek2"},"attachments":[{"id":98212637,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/98212637/thumbnails/1.jpg","file_name":"61c541c2a67ae67c82ee1f2217032b43102b.pdf","download_url":"https://www.academia.edu/attachments/98212637/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"The_actions_on_the_generating_functions.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/98212637/61c541c2a67ae67c82ee1f2217032b43102b-libre.pdf?1675494178=\u0026response-content-disposition=attachment%3B+filename%3DThe_actions_on_the_generating_functions.pdf\u0026Expires=1733183071\u0026Signature=QMn16a9Q6OBtVsehFz2Dwl2CULtawLsj0XGc5~uDw0eleOTsyjMovvPkgYuz2NwRKafkqm1TeyEJsVd7nVhlrAw9wWRYpRwqhmB9l0pgKTR7dSIRBw1HCeR5zRAC~c41oxFlLxhT1T47xXHGGUzc2bygDDmFo1PZjoUcP9kIdO01ZLhSJxk-kyX1I0~YCqQ5QmNMOhN9LIrfZuOWMzxeK9E07UEW1pewYUjZLoAongfkPGMjxMOcuwvxQbihxlFj8xS~ZgE~Q2HZSPvSLeIQgHdcDGaNWjQ06-ssFesJALUSE56kvEERseXOY3rdxcJNBBlODxAAyUOqpGCiZ97xZQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":305,"name":"Applied Mathematics","url":"https://www.academia.edu/Documents/in/Applied_Mathematics"},{"id":54284,"name":"Generalization","url":"https://www.academia.edu/Documents/in/Generalization"},{"id":3236149,"name":"Homomorphism","url":"https://www.academia.edu/Documents/in/Homomorphism"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="89898484"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/89898484/A_special_approach_to_derive_new_formulas_for_some_special_numbers_and_polynomials"><img alt="Research paper thumbnail of A special approach to derive new formulas for some special numbers and polynomials" class="work-thumbnail" src="https://attachments.academia-assets.com/93611581/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/89898484/A_special_approach_to_derive_new_formulas_for_some_special_numbers_and_polynomials">A special approach to derive new formulas for some special numbers and polynomials</a></div><div class="wp-workCard_item"><span>TURKISH JOURNAL OF MATHEMATICS</span><span>, 2020</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="8e272ba61afa9fe099ab005de1bc4a25" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":93611581,"asset_id":89898484,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/93611581/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="89898484"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="89898484"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 89898484; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=89898484]").text(description); $(".js-view-count[data-work-id=89898484]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 89898484; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='89898484']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 89898484, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); 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Next, using the Legendre identity, trigonometric functions and the Dirichlet kernel, some formulas and relations involving Bernoulli and Euler numbers, cosine-type Bernoulli and Euler polynomials, and sine-type Bernoulli and Euler polynomials are driven. Then, by using the generating functions method and the well-known Euler identity, many new identities, formulas, and combinatorial sums among the Fibonacci numbers and polynomials, the Lucas numbers and polynomials, the Chebyshev polynomials, and Bernoulli and Euler type polynomials are given. Finally, some infinite series representations for these special numbers and polynomials and their numerical examples are presented.","publication_date":{"day":null,"month":null,"year":2020,"errors":{}},"publication_name":"TURKISH JOURNAL OF MATHEMATICS","grobid_abstract_attachment_id":93611581},"translated_abstract":null,"internal_url":"https://www.academia.edu/89898484/A_special_approach_to_derive_new_formulas_for_some_special_numbers_and_polynomials","translated_internal_url":"","created_at":"2022-11-03T12:28:10.428-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":146832572,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":93611581,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/93611581/thumbnails/1.jpg","file_name":"openAcceptedDocument.pdf","download_url":"https://www.academia.edu/attachments/93611581/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"A_special_approach_to_derive_new_formula.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/93611581/openAcceptedDocument-libre.pdf?1667506642=\u0026response-content-disposition=attachment%3B+filename%3DA_special_approach_to_derive_new_formula.pdf\u0026Expires=1733183071\u0026Signature=Mm5HprYrHEwbH379GpKngqR0XK37Dv1PH7THhd3UoW8J33~7Nf9MI~u0mumk1zmiSC77oOlGlCLQ7L7WH7~A91LvaIB2dO9zrGIK-dNf92tl8~BcBNNf7yoLMKeWAdZ46no0G3ko92rjxyjjJjUGNxgl7UEX0-YBeDSDY3VfIFRBER12dzcs582hRluzI~~g7EVP0Ei5X~xNMJZTloMgldfDCrHBltTSLHxgN54Qr3JdEYgZrKOJz-RGG9R8tXhxZqTF6d9HZxfuycWtyWtTVVsH305PTtYyDbV2j3-~xXWdjrlXzim2M5BPfKHEQNIj6-nn5keD2HJNnI8Zh0AXUQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"A_special_approach_to_derive_new_formulas_for_some_special_numbers_and_polynomials","translated_slug":"","page_count":24,"language":"en","content_type":"Work","owner":{"id":146832572,"first_name":"Yilmaz","middle_initials":null,"last_name":"Simsek","page_name":"YilmazSimsek2","domain_name":"independent","created_at":"2020-02-22T04:58:30.104-08:00","display_name":"Yilmaz Simsek","url":"https://independent.academia.edu/YilmazSimsek2"},"attachments":[{"id":93611581,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/93611581/thumbnails/1.jpg","file_name":"openAcceptedDocument.pdf","download_url":"https://www.academia.edu/attachments/93611581/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"A_special_approach_to_derive_new_formula.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/93611581/openAcceptedDocument-libre.pdf?1667506642=\u0026response-content-disposition=attachment%3B+filename%3DA_special_approach_to_derive_new_formula.pdf\u0026Expires=1733183071\u0026Signature=Mm5HprYrHEwbH379GpKngqR0XK37Dv1PH7THhd3UoW8J33~7Nf9MI~u0mumk1zmiSC77oOlGlCLQ7L7WH7~A91LvaIB2dO9zrGIK-dNf92tl8~BcBNNf7yoLMKeWAdZ46no0G3ko92rjxyjjJjUGNxgl7UEX0-YBeDSDY3VfIFRBER12dzcs582hRluzI~~g7EVP0Ei5X~xNMJZTloMgldfDCrHBltTSLHxgN54Qr3JdEYgZrKOJz-RGG9R8tXhxZqTF6d9HZxfuycWtyWtTVVsH305PTtYyDbV2j3-~xXWdjrlXzim2M5BPfKHEQNIj6-nn5keD2HJNnI8Zh0AXUQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="81374464"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/81374464/Identities_for_Dirichlet_and_Lambert_type_series_arising_from_the_numbers_of_a_certain_special_word"><img alt="Research paper thumbnail of Identities for Dirichlet and Lambert-type series arising from the numbers of a certain special word" class="work-thumbnail" src="https://attachments.academia-assets.com/87438925/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/81374464/Identities_for_Dirichlet_and_Lambert_type_series_arising_from_the_numbers_of_a_certain_special_word">Identities for Dirichlet and Lambert-type series arising from the numbers of a certain special word</a></div><div class="wp-workCard_item"><span>Applicable Analysis and Discrete Mathematics</span><span>, 2019</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The goal of this paper is to give several new Dirichlet-type series associated with the Riemann z...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The goal of this paper is to give several new Dirichlet-type series associated with the Riemann zeta function, the polylogarithm function, and also the numbers of necklaces and Lyndon words. By applying Dirichlet convolution formula to number-theoretic functions related to these series, various novel identities and relations are derived. Moreover, some new formulas related to Bernoulli-type numbers and polynomials obtain from generating functions and these Dirichlet-type series. Finally, several relations among the Fourier expansion of Eisenstein series, the Lambert series and the number-theoretic functions are given.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="0a39c850afe2440d093988181f072683" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":87438925,"asset_id":81374464,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/87438925/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="81374464"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="81374464"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 81374464; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=81374464]").text(description); $(".js-view-count[data-work-id=81374464]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 81374464; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='81374464']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 81374464, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "0a39c850afe2440d093988181f072683" } } $('.js-work-strip[data-work-id=81374464]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":81374464,"title":"Identities for Dirichlet and Lambert-type series arising from the numbers of a certain special word","translated_title":"","metadata":{"abstract":"The goal of this paper is to give several new Dirichlet-type series associated with the Riemann zeta function, the polylogarithm function, and also the numbers of necklaces and Lyndon words. By applying Dirichlet convolution formula to number-theoretic functions related to these series, various novel identities and relations are derived. Moreover, some new formulas related to Bernoulli-type numbers and polynomials obtain from generating functions and these Dirichlet-type series. Finally, several relations among the Fourier expansion of Eisenstein series, the Lambert series and the number-theoretic functions are given.","publisher":"National Library of Serbia","publication_date":{"day":null,"month":null,"year":2019,"errors":{}},"publication_name":"Applicable Analysis and Discrete Mathematics"},"translated_abstract":"The goal of this paper is to give several new Dirichlet-type series associated with the Riemann zeta function, the polylogarithm function, and also the numbers of necklaces and Lyndon words. By applying Dirichlet convolution formula to number-theoretic functions related to these series, various novel identities and relations are derived. Moreover, some new formulas related to Bernoulli-type numbers and polynomials obtain from generating functions and these Dirichlet-type series. Finally, several relations among the Fourier expansion of Eisenstein series, the Lambert series and the number-theoretic functions are given.","internal_url":"https://www.academia.edu/81374464/Identities_for_Dirichlet_and_Lambert_type_series_arising_from_the_numbers_of_a_certain_special_word","translated_internal_url":"","created_at":"2022-06-13T04:03:09.837-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":146832572,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":87438925,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/87438925/thumbnails/1.jpg","file_name":"1452-86301900033K.pdf","download_url":"https://www.academia.edu/attachments/87438925/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Identities_for_Dirichlet_and_Lambert_typ.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/87438925/1452-86301900033K-libre.pdf?1655118710=\u0026response-content-disposition=attachment%3B+filename%3DIdentities_for_Dirichlet_and_Lambert_typ.pdf\u0026Expires=1733183071\u0026Signature=Vr4DEmX6QRJxLLQGzxPFA0uBTkTHir4PP1LnMEFxxHjukQptPdC1vATTI0f1-5EzQlTKgIblp6Qbp5m58szWvrolyMOsimnkkiZiI5gFdyYa5d5cshpmrpZdL-2y51qp6cvz7O9bDgvcV8H1o2A7ItccrOTIhWQUowI-gKSVSv9ENuMOW~VJIJIyn-p0nvwrIzqOA-8d1GTUKXSvKnjLvKLM39jYSDMsEnTDnR24UZl6aYdY3LH8vag9O4Y9VNVcRGlZwU2G7XVjczb6xyEeIRzy3i877-AyJxrjskLGcx11iLrsl2u-kgBCBRL0N5u4DDaS7~H4yEWgQJzDLkcfcA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Identities_for_Dirichlet_and_Lambert_type_series_arising_from_the_numbers_of_a_certain_special_word","translated_slug":"","page_count":18,"language":"en","content_type":"Work","owner":{"id":146832572,"first_name":"Yilmaz","middle_initials":null,"last_name":"Simsek","page_name":"YilmazSimsek2","domain_name":"independent","created_at":"2020-02-22T04:58:30.104-08:00","display_name":"Yilmaz Simsek","url":"https://independent.academia.edu/YilmazSimsek2"},"attachments":[{"id":87438925,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/87438925/thumbnails/1.jpg","file_name":"1452-86301900033K.pdf","download_url":"https://www.academia.edu/attachments/87438925/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Identities_for_Dirichlet_and_Lambert_typ.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/87438925/1452-86301900033K-libre.pdf?1655118710=\u0026response-content-disposition=attachment%3B+filename%3DIdentities_for_Dirichlet_and_Lambert_typ.pdf\u0026Expires=1733183071\u0026Signature=Vr4DEmX6QRJxLLQGzxPFA0uBTkTHir4PP1LnMEFxxHjukQptPdC1vATTI0f1-5EzQlTKgIblp6Qbp5m58szWvrolyMOsimnkkiZiI5gFdyYa5d5cshpmrpZdL-2y51qp6cvz7O9bDgvcV8H1o2A7ItccrOTIhWQUowI-gKSVSv9ENuMOW~VJIJIyn-p0nvwrIzqOA-8d1GTUKXSvKnjLvKLM39jYSDMsEnTDnR24UZl6aYdY3LH8vag9O4Y9VNVcRGlZwU2G7XVjczb6xyEeIRzy3i877-AyJxrjskLGcx11iLrsl2u-kgBCBRL0N5u4DDaS7~H4yEWgQJzDLkcfcA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":556845,"name":"Numerical Analysis and Computational Mathematics","url":"https://www.academia.edu/Documents/in/Numerical_Analysis_and_Computational_Mathematics"},{"id":3777346,"name":"Dirichlet series","url":"https://www.academia.edu/Documents/in/Dirichlet_series"}],"urls":[]}, dispatcherData: dispatcherData }); 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="77100442"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/77100442/On_Generating_Functions_for_Parametrically_Generalized_Polynomials_Involving_Combinatorial_Bernoulli_and_Euler_Polynomials_and_Numbers"><img alt="Research paper thumbnail of On Generating Functions for Parametrically Generalized Polynomials Involving Combinatorial, Bernoulli and Euler Polynomials and Numbers" class="work-thumbnail" src="https://attachments.academia-assets.com/84559082/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" 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><div class="wp-workCard_item"><span>Symmetry</span><span>, 2022</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The aim of this paper is to give generating functions for parametrically generalized polynomials ...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The aim of this paper is to give generating functions for parametrically generalized polynomials that are related to the combinatorial numbers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the cosine-Bernoulli polynomials, the sine-Bernoulli polynomials, the cosine-Euler polynomials, and the sine-Euler polynomials. We investigate some properties of these generating functions. By applying Euler’s formula to these generating functions, we derive many new and interesting formulas and relations related to these special polynomials and numbers mentioned as above. Some special cases of the results obtained in this article are examined. With this special case, detailed comments and comparisons with previously available results are also provided. Furthermore, we come up with open questions about interpolation functions for these polynomials. The main results of this paper highlight the existing symmetry between numbers and polynomials in a more general framework...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="0d037e844b355f646fe9070914d6c26a" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":84559082,"asset_id":77100442,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/84559082/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77100442"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77100442"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77100442; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77100442]").text(description); $(".js-view-count[data-work-id=77100442]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77100442; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77100442']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77100442, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "0d037e844b355f646fe9070914d6c26a" } } $('.js-work-strip[data-work-id=77100442]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77100442,"title":"On Generating Functions for Parametrically Generalized Polynomials Involving Combinatorial, Bernoulli and Euler Polynomials and Numbers","translated_title":"","metadata":{"abstract":"The aim of this paper is to give generating functions for parametrically generalized polynomials that are related to the combinatorial numbers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the cosine-Bernoulli polynomials, the sine-Bernoulli polynomials, the cosine-Euler polynomials, and the sine-Euler polynomials. We investigate some properties of these generating functions. By applying Euler’s formula to these generating functions, we derive many new and interesting formulas and relations related to these special polynomials and numbers mentioned as above. Some special cases of the results obtained in this article are examined. With this special case, detailed comments and comparisons with previously available results are also provided. Furthermore, we come up with open questions about interpolation functions for these polynomials. The main results of this paper highlight the existing symmetry between numbers and polynomials in a more general framework...","publisher":"MDPI AG","publication_date":{"day":null,"month":null,"year":2022,"errors":{}},"publication_name":"Symmetry"},"translated_abstract":"The aim of this paper is to give generating functions for parametrically generalized polynomials that are related to the combinatorial numbers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the cosine-Bernoulli polynomials, the sine-Bernoulli polynomials, the cosine-Euler polynomials, and the sine-Euler polynomials. We investigate some properties of these generating functions. By applying Euler’s formula to these generating functions, we derive many new and interesting formulas and relations related to these special polynomials and numbers mentioned as above. Some special cases of the results obtained in this article are examined. 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The main results of this paper highlight the existing symmetry between numbers and polynomials in a more general framework...","internal_url":"https://www.academia.edu/77100442/On_Generating_Functions_for_Parametrically_Generalized_Polynomials_Involving_Combinatorial_Bernoulli_and_Euler_Polynomials_and_Numbers","translated_internal_url":"","created_at":"2022-04-20T11:12:48.092-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":146832572,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":84559082,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559082/thumbnails/1.jpg","file_name":"pdf.pdf","download_url":"https://www.academia.edu/attachments/84559082/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_Generating_Functions_for_Parametrical.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559082/pdf-libre.pdf?1650478773=\u0026response-content-disposition=attachment%3B+filename%3DOn_Generating_Functions_for_Parametrical.pdf\u0026Expires=1733183071\u0026Signature=XAnw-zkJfw~6OB17TCkeDY7hdV3KtfVAuiCDJd~wSzfHnTnUn3d5wtyaCLXMJFzU~qsiSXeS6vMM-WTOlMON2QegbRVqQFeN4F0LL9qAD0AtPvPmQOzPY~hukFmGqpPH7-2HWpJc8auCZgE0e73v-4L9LdhNVcr~wGZg0Vgjbkrys6OwH2NysEWRYoZIuOJprE-0sgtcXxlBTRMZHyLUVXabYEB0bnej3nbQ2wRDwd~07Joh1qk6clx07RFDx9sgHiZBjCyreze1jm2Ju7DEo-2mDp-IFYtA5cmB8NYRepj1baIGPraSy~O6W6D--BrBJyB2DZD8qRa5W4ibD8MvPA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"On_Generating_Functions_for_Parametrically_Generalized_Polynomials_Involving_Combinatorial_Bernoulli_and_Euler_Polynomials_and_Numbers","translated_slug":"","page_count":12,"language":"en","content_type":"Work","owner":{"id":146832572,"first_name":"Yilmaz","middle_initials":null,"last_name":"Simsek","page_name":"YilmazSimsek2","domain_name":"independent","created_at":"2020-02-22T04:58:30.104-08:00","display_name":"Yilmaz Simsek","url":"https://independent.academia.edu/YilmazSimsek2"},"attachments":[{"id":84559082,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559082/thumbnails/1.jpg","file_name":"pdf.pdf","download_url":"https://www.academia.edu/attachments/84559082/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"On_Generating_Functions_for_Parametrical.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559082/pdf-libre.pdf?1650478773=\u0026response-content-disposition=attachment%3B+filename%3DOn_Generating_Functions_for_Parametrical.pdf\u0026Expires=1733183071\u0026Signature=XAnw-zkJfw~6OB17TCkeDY7hdV3KtfVAuiCDJd~wSzfHnTnUn3d5wtyaCLXMJFzU~qsiSXeS6vMM-WTOlMON2QegbRVqQFeN4F0LL9qAD0AtPvPmQOzPY~hukFmGqpPH7-2HWpJc8auCZgE0e73v-4L9LdhNVcr~wGZg0Vgjbkrys6OwH2NysEWRYoZIuOJprE-0sgtcXxlBTRMZHyLUVXabYEB0bnej3nbQ2wRDwd~07Joh1qk6clx07RFDx9sgHiZBjCyreze1jm2Ju7DEo-2mDp-IFYtA5cmB8NYRepj1baIGPraSy~O6W6D--BrBJyB2DZD8qRa5W4ibD8MvPA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":126829,"name":"Symmetry","url":"https://www.academia.edu/Documents/in/Symmetry"}],"urls":[{"id":19721510,"url":"https://www.mdpi.com/2073-8994/14/4/654/pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="77100441"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" rel="nofollow" href="https://www.academia.edu/77100441/Formulas_for_characteristic_function_and_moment_generating_functions_of_beta_type_distribution"><img alt="Research paper thumbnail of Formulas for characteristic function and moment generating functions of beta type distribution" class="work-thumbnail" src="https://a.academia-assets.com/images/blank-paper.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" rel="nofollow" href="https://www.academia.edu/77100441/Formulas_for_characteristic_function_and_moment_generating_functions_of_beta_type_distribution">Formulas for characteristic function and moment generating functions of beta type distribution</a></div><div class="wp-workCard_item"><span>Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas</span><span>, 2022</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77100441"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77100441"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77100441; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77100441]").text(description); $(".js-view-count[data-work-id=77100441]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77100441; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77100441']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77100441, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (false){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "-1" } } $('.js-work-strip[data-work-id=77100441]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77100441,"title":"Formulas for characteristic function and moment generating functions of beta type distribution","translated_title":"","metadata":{"publisher":"Springer Science and Business Media LLC","publication_date":{"day":null,"month":null,"year":2022,"errors":{}},"publication_name":"Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. 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Reciprocity laws for these sums are proven. Some applications of these reciprocity laws are presented. With aid of the reciprocity law of the Dedekind sums, formulas for many new finite sums are obtained. Relations among these new classes of finite sums, partial sum of the generalized harmonic functions, the Riemann zeta function, the Hurwitz zeta function, hypergeometric series, polylogarithms, digamma functions, polygmamma functions, and special numbers and polynomials and other combinatorial sums are given. Moreover, some formulas for the partial sum of the generalized harmonic functions and special numbers and polynomials are given. Finally, coments and observations on the results of this paper are given.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2b2527cedbf392f40d931058f88f1e08" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":84559080,"asset_id":77100440,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/84559080/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77100440"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77100440"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77100440; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77100440]").text(description); $(".js-view-count[data-work-id=77100440]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77100440; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77100440']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77100440, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "2b2527cedbf392f40d931058f88f1e08" } } $('.js-work-strip[data-work-id=77100440]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77100440,"title":"Some Classes of Finite Sums Related to the Generalized Harmonic Functions and Special Numbers and Polynomials","translated_title":"","metadata":{"abstract":"The aim of this paper is to give some new classes of finite sums involving the numbers y (m, λ), the generalized harmonic functions, special numbers and polynomials, the Dedekind sums, and other combinatorial sum. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="77100439"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/77100439/Computation_Methods_for_combinatorial_sums_and_Euler_type_numbers_related_to_new_families_of_numbers"><img alt="Research paper thumbnail of Computation Methods for combinatorial sums and Euler type numbers related to new families of numbers" class="work-thumbnail" src="https://attachments.academia-assets.com/84559078/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/77100439/Computation_Methods_for_combinatorial_sums_and_Euler_type_numbers_related_to_new_families_of_numbers">Computation Methods for combinatorial sums and Euler type numbers related to new families of numbers</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The aim of this article is to define some new families of the special numbers. These numbers prov...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order. We can show that these numbers are related to the well-known numbers and polynomials such as the Stirling numbers of the second kind and the central factorial numbers, the array polynomials, the rook numbers and polynomials, the Bernstein basis functions and others. In order to derive our new identities and relations for these numbers, we use a technique including the generating functions and functional equations. Finally, we not only give a computational algorithm for these numbers, but also some numerical values of these numbers and the Euler numbers of negative order with tables. We also give some combinatorial interpretations of our new numbers.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="8bc2052e03bc43a6c807b49282448e3f" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":84559078,"asset_id":77100439,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/84559078/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77100439"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77100439"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77100439; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77100439]").text(description); $(".js-view-count[data-work-id=77100439]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77100439; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77100439']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77100439, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "8bc2052e03bc43a6c807b49282448e3f" } } $('.js-work-strip[data-work-id=77100439]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77100439,"title":"Computation Methods for combinatorial sums and Euler type numbers related to new families of numbers","translated_title":"","metadata":{"abstract":"The aim of this article is to define some new families of the special numbers. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="77100438"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/77100438/New_families_of_special_numbers_for_computing_negative_order_Euler_numbers"><img alt="Research paper thumbnail of New families of special numbers for computing negative order Euler numbers" class="work-thumbnail" src="https://attachments.academia-assets.com/84559077/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/77100438/New_families_of_special_numbers_for_computing_negative_order_Euler_numbers">New families of special numbers for computing negative order Euler numbers</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The main purpose of this paper is to construct new families of special numbers with their generat...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The main purpose of this paper is to construct new families of special numbers with their generating functions. These numbers are related to the many well-known numbers, which are the Bernoulli numbers, the Fibonacci numbers, the Lucas numbers, the Stirling numbers of the second kind and the central factorial numbers. Our other inspiration of this paper is related to the Golombek&#39;s problem golombek“ Aufgabe 1088, El. Math. 49 (1994) 126-127” . Our first numbers are not only related to the Golombek&#39;s problem, but also computation of the negative order Euler numbers. We compute a few values of the numbers which are given by some tables. We give some applications in Probability and Statistics. That is, special values of mathematical expectation in the binomial distribution and the Bernstein polynomials give us the value of our numbers. Taking derivative of our generating functions, we give partial differential equations and also functional equations. By using these equations, w...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="096acc50d8cea275a4f1fd9b6e08bf75" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":84559077,"asset_id":77100438,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/84559077/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77100438"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77100438"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77100438; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77100438]").text(description); $(".js-view-count[data-work-id=77100438]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77100438; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77100438']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77100438, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "096acc50d8cea275a4f1fd9b6e08bf75" } } $('.js-work-strip[data-work-id=77100438]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77100438,"title":"New families of special numbers for computing negative order Euler numbers","translated_title":"","metadata":{"abstract":"The main purpose of this paper is to construct new families of special numbers with their generating functions. These numbers are related to the many well-known numbers, which are the Bernoulli numbers, the Fibonacci numbers, the Lucas numbers, the Stirling numbers of the second kind and the central factorial numbers. Our other inspiration of this paper is related to the Golombek\u0026#39;s problem golombek“ Aufgabe 1088, El. Math. 49 (1994) 126-127” . Our first numbers are not only related to the Golombek\u0026#39;s problem, but also computation of the negative order Euler numbers. We compute a few values of the numbers which are given by some tables. We give some applications in Probability and Statistics. That is, special values of mathematical expectation in the binomial distribution and the Bernstein polynomials give us the value of our numbers. Taking derivative of our generating functions, we give partial differential equations and also functional equations. By using these equations, w...","publication_date":{"day":19,"month":4,"year":2016,"errors":{}}},"translated_abstract":"The main purpose of this paper is to construct new families of special numbers with their generating functions. These numbers are related to the many well-known numbers, which are the Bernoulli numbers, the Fibonacci numbers, the Lucas numbers, the Stirling numbers of the second kind and the central factorial numbers. Our other inspiration of this paper is related to the Golombek\u0026#39;s problem golombek“ Aufgabe 1088, El. Math. 49 (1994) 126-127” . Our first numbers are not only related to the Golombek\u0026#39;s problem, but also computation of the negative order Euler numbers. We compute a few values of the numbers which are given by some tables. We give some applications in Probability and Statistics. That is, special values of mathematical expectation in the binomial distribution and the Bernstein polynomials give us the value of our numbers. Taking derivative of our generating functions, we give partial differential equations and also functional equations. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="77100436"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/77100436/Derivation_of_Computational_Formulas_for_certain_class_of_finite_sums_Approach_to_Generating_functions_arising_from_p_adic_integrals_and_special_functions"><img alt="Research paper thumbnail of Derivation of Computational Formulas for certain class of finite sums: Approach to Generating functions arising from p-adic integrals and special functions" class="work-thumbnail" src="https://attachments.academia-assets.com/84559074/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/77100436/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><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The aim of this paper is to construct generating functions for some families of special finite su...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">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...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="4f66cb14d667113a8c6d25619f7822ee" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":84559074,"asset_id":77100436,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/84559074/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77100436"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77100436"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77100436; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77100436]").text(description); $(".js-view-count[data-work-id=77100436]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77100436; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77100436']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77100436, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "4f66cb14d667113a8c6d25619f7822ee" } } $('.js-work-strip[data-work-id=77100436]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77100436,"title":"Derivation of Computational Formulas for certain class of finite sums: Approach to Generating functions arising from p-adic integrals and special functions","translated_title":"","metadata":{"abstract":"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). 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Morover, using the aforementioned special finite sums and combinatorial numbers, we give relations among multiple alte...","publication_date":{"day":22,"month":8,"year":2021,"errors":{}}},"translated_abstract":"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...","internal_url":"https://www.academia.edu/77100436/Derivation_of_Computational_Formulas_for_certain_class_of_finite_sums_Approach_to_Generating_functions_arising_from_p_adic_integrals_and_special_functions","translated_internal_url":"","created_at":"2022-04-20T11:12:46.847-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":146832572,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":84559074,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559074/thumbnails/1.jpg","file_name":"2108.10756v1.pdf","download_url":"https://www.academia.edu/attachments/84559074/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Derivation_of_Computational_Formulas_for.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559074/2108.10756v1-libre.pdf?1650478777=\u0026response-content-disposition=attachment%3B+filename%3DDerivation_of_Computational_Formulas_for.pdf\u0026Expires=1733183071\u0026Signature=Ml1sYJag5F8G3lD7~WJiKIsMRbm1~3mxOGgteDGXk3rGsaI1tZI291~3BzliPjiu4hvaeOjmJuN8jtMvo3dDvlcivOOfFj0fNGf98FVK5MZHSbphd1rAezj92PxpsCQQRRY4nQrbVw72HEVh44kVTXqYO-AGlJnuH~4iF9OYxNlEQji61o9-UMS-m7K4LVR1lB1kv5F8gcUoR4711q1KahiXOFdudAbxcdIfgR0rdudz95j3j8kELWvMg71GA90Jf2vKVX~Ayua-D2W2hTlAkBc8LVP8CarXUYnUtqrd7Oom986zpcfAjLiBF8j7oP1I~jqPf8P8Byd2x-hVrAXJfg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Derivation_of_Computational_Formulas_for_certain_class_of_finite_sums_Approach_to_Generating_functions_arising_from_p_adic_integrals_and_special_functions","translated_slug":"","page_count":33,"language":"en","content_type":"Work","owner":{"id":146832572,"first_name":"Yilmaz","middle_initials":null,"last_name":"Simsek","page_name":"YilmazSimsek2","domain_name":"independent","created_at":"2020-02-22T04:58:30.104-08:00","display_name":"Yilmaz Simsek","url":"https://independent.academia.edu/YilmazSimsek2"},"attachments":[{"id":84559074,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559074/thumbnails/1.jpg","file_name":"2108.10756v1.pdf","download_url":"https://www.academia.edu/attachments/84559074/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Derivation_of_Computational_Formulas_for.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559074/2108.10756v1-libre.pdf?1650478777=\u0026response-content-disposition=attachment%3B+filename%3DDerivation_of_Computational_Formulas_for.pdf\u0026Expires=1733183071\u0026Signature=Ml1sYJag5F8G3lD7~WJiKIsMRbm1~3mxOGgteDGXk3rGsaI1tZI291~3BzliPjiu4hvaeOjmJuN8jtMvo3dDvlcivOOfFj0fNGf98FVK5MZHSbphd1rAezj92PxpsCQQRRY4nQrbVw72HEVh44kVTXqYO-AGlJnuH~4iF9OYxNlEQji61o9-UMS-m7K4LVR1lB1kv5F8gcUoR4711q1KahiXOFdudAbxcdIfgR0rdudz95j3j8kELWvMg71GA90Jf2vKVX~Ayua-D2W2hTlAkBc8LVP8CarXUYnUtqrd7Oom986zpcfAjLiBF8j7oP1I~jqPf8P8Byd2x-hVrAXJfg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":84559075,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559075/thumbnails/1.jpg","file_name":"2108.10756v1.pdf","download_url":"https://www.academia.edu/attachments/84559075/download_file","bulk_download_file_name":"Derivation_of_Computational_Formulas_for.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559075/2108.10756v1-libre.pdf?1650478778=\u0026response-content-disposition=attachment%3B+filename%3DDerivation_of_Computational_Formulas_for.pdf\u0026Expires=1733183071\u0026Signature=E6iUIU6j1XeMFazDDrS-icMY9RMYRJv7yPdo7K6TTnQKy94MGSEJA65qdCS6lJaFTn8XN8CtP4KLGZGOgZi61RrQFs86-vnFAyQSMtK5-MvS62tEexvb-NSCazGaofxj~hTYNs9LANX9kQmmgmdkCGg~RALMmG1ty6W0HKB2xowQdTUBkdvNgTl8oGpj9eJTJcPD4bPeTeNXLVkFEgmPxGz02MhOUYRn-OmEm1Uj72mgWv-iv8slVuuj0PiO3Cnlv20LvJ9ZxnslQ-ZnD-Oi1nNp5bXn2nGtFayCaYDTakY5ArbnQ1BeLamr28UncN0z5iyNPB0yzV1Qo9ODY0Oy9g__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"}],"urls":[{"id":19721505,"url":"https://arxiv.org/pdf/2108.10756v1.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="77100435"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/77100435/A_p_adic_look_at_the_Diophantine_equation_x_2_11_2k_y_n"><img alt="Research paper thumbnail of A p-adic look at the Diophantine equation x^2+11^2k=y^n" class="work-thumbnail" src="https://attachments.academia-assets.com/84954988/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/77100435/A_p_adic_look_at_the_Diophantine_equation_x_2_11_2k_y_n">A p-adic look at the Diophantine equation x^2+11^2k=y^n</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We find all solutions of Diophantine equation x^2+11^2k = y^n where x&gt;=1, y&gt;=1, n&gt;=3 and...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We find all solutions of Diophantine equation x^2+11^2k = y^n where x&gt;=1, y&gt;=1, n&gt;=3 and k is natural number. We give p-adic interpretation of this equation.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="ea6d3901a0adc49dd2b98eeaa4648f79" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":84954988,"asset_id":77100435,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/84954988/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77100435"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77100435"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77100435; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77100435]").text(description); $(".js-view-count[data-work-id=77100435]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77100435; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77100435']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77100435, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "ea6d3901a0adc49dd2b98eeaa4648f79" } } $('.js-work-strip[data-work-id=77100435]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77100435,"title":"A p-adic look at the Diophantine equation x^2+11^2k=y^n","translated_title":"","metadata":{"abstract":"We find all solutions of Diophantine equation x^2+11^2k = y^n where x\u0026gt;=1, y\u0026gt;=1, n\u0026gt;=3 and k is natural number. We give p-adic interpretation of this equation.","publication_date":{"day":27,"month":12,"year":2011,"errors":{}}},"translated_abstract":"We find all solutions of Diophantine equation x^2+11^2k = y^n where x\u0026gt;=1, y\u0026gt;=1, n\u0026gt;=3 and k is natural number. We give p-adic interpretation of this equation.","internal_url":"https://www.academia.edu/77100435/A_p_adic_look_at_the_Diophantine_equation_x_2_11_2k_y_n","translated_internal_url":"","created_at":"2022-04-20T11:12:46.602-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":146832572,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":84954988,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84954988/thumbnails/1.jpg","file_name":"1112.pdf","download_url":"https://www.academia.edu/attachments/84954988/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"A_p_adic_look_at_the_Diophantine_equatio.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84954988/1112-libre.pdf?1650959490=\u0026response-content-disposition=attachment%3B+filename%3DA_p_adic_look_at_the_Diophantine_equatio.pdf\u0026Expires=1733183071\u0026Signature=YJ4KkVT6pM5jVLxEZJLk3dndmoGXR0daUObakYwU9eZrFvW9tmTwU37c6ftSMgadTjSCJC0mWwZg8TvtgMmu7Anafy~YLJ2X5jFFZVRVTaI1cTZTJ75JpRIWVedjg8sNfgMBYaWOpA2SiShayk-Z~cqh83zQrduFLOlfqAC7JLQqYp~DqrWJK42nGDINLT~JMxXnqFTrCVKwug7YfkdnV~PLryofjw0~kYG~fzlIBkdgrPgkFVTBRvv3toOAqyPewo4kKafVa8fJB67IK4hztOINOHy9w3ZLlsh~1ocFLjAfx7UQ8ElVKwR6ZAJ2S2AO7RggmRRPT73Pf2vruYKz6Q__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"A_p_adic_look_at_the_Diophantine_equation_x_2_11_2k_y_n","translated_slug":"","page_count":4,"language":"en","content_type":"Work","owner":{"id":146832572,"first_name":"Yilmaz","middle_initials":null,"last_name":"Simsek","page_name":"YilmazSimsek2","domain_name":"independent","created_at":"2020-02-22T04:58:30.104-08:00","display_name":"Yilmaz Simsek","url":"https://independent.academia.edu/YilmazSimsek2"},"attachments":[{"id":84954988,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84954988/thumbnails/1.jpg","file_name":"1112.pdf","download_url":"https://www.academia.edu/attachments/84954988/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"A_p_adic_look_at_the_Diophantine_equatio.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84954988/1112-libre.pdf?1650959490=\u0026response-content-disposition=attachment%3B+filename%3DA_p_adic_look_at_the_Diophantine_equatio.pdf\u0026Expires=1733183071\u0026Signature=YJ4KkVT6pM5jVLxEZJLk3dndmoGXR0daUObakYwU9eZrFvW9tmTwU37c6ftSMgadTjSCJC0mWwZg8TvtgMmu7Anafy~YLJ2X5jFFZVRVTaI1cTZTJ75JpRIWVedjg8sNfgMBYaWOpA2SiShayk-Z~cqh83zQrduFLOlfqAC7JLQqYp~DqrWJK42nGDINLT~JMxXnqFTrCVKwug7YfkdnV~PLryofjw0~kYG~fzlIBkdgrPgkFVTBRvv3toOAqyPewo4kKafVa8fJB67IK4hztOINOHy9w3ZLlsh~1ocFLjAfx7UQ8ElVKwR6ZAJ2S2AO7RggmRRPT73Pf2vruYKz6Q__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":194130,"name":"PARTIAL DIFFERENTIAL EQUATION","url":"https://www.academia.edu/Documents/in/PARTIAL_DIFFERENTIAL_EQUATION"},{"id":506482,"name":"Function approximation","url":"https://www.academia.edu/Documents/in/Function_approximation"},{"id":939537,"name":"Boundary Value Problem","url":"https://www.academia.edu/Documents/in/Boundary_Value_Problem"},{"id":3641544,"name":"Diophantine Equation","url":"https://www.academia.edu/Documents/in/Diophantine_Equation"}],"urls":[{"id":19721504,"url":"https://core.ac.uk/download/pdf/2244862.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="77100434"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/77100434/Identities_and_relations_related_to_the_numbers_of_special_words_derived_from_special_series_with_Dirichlet_convolution"><img alt="Research paper thumbnail of Identities and relations related to the numbers of special words derived from special series with Dirichlet convolution" class="work-thumbnail" src="https://attachments.academia-assets.com/84559072/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/77100434/Identities_and_relations_related_to_the_numbers_of_special_words_derived_from_special_series_with_Dirichlet_convolution">Identities and relations related to the numbers of special words derived from special series with Dirichlet convolution</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The aim of this paper is to define some new number-theoretic functions including necklaces polyno...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The aim of this paper is to define some new number-theoretic functions including necklaces polynomials and the numbers of special words such as Lyndon words. By using Dirichlet convolution formula with well-known number-theoretic functions, we derive some new identities and relations associated with Dirichlet series, Lambert series, and also the family of zeta functions including the Riemann zeta functions and polylogarithm functions. By using analytic (meromorphic) continuation of zeta functions, we also derive identities and formulas including Bernoulli numbers and Apostol-Bernoulli numbers. Moreover, we give relations between number-theoretic functions and the Fourier expansion of the Eisenstein series. Finally, we give some observations and remarks on these functions.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="418c97e8a16bc65033893e247f271285" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":84559072,"asset_id":77100434,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/84559072/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77100434"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77100434"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77100434; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77100434]").text(description); $(".js-view-count[data-work-id=77100434]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77100434; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77100434']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77100434, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "418c97e8a16bc65033893e247f271285" } } $('.js-work-strip[data-work-id=77100434]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77100434,"title":"Identities and relations related to the numbers of special words derived from special series with Dirichlet convolution","translated_title":"","metadata":{"abstract":"The aim of this paper is to define some new number-theoretic functions including necklaces polynomials and the numbers of special words such as Lyndon words. By using Dirichlet convolution formula with well-known number-theoretic functions, we derive some new identities and relations associated with Dirichlet series, Lambert series, and also the family of zeta functions including the Riemann zeta functions and polylogarithm functions. By using analytic (meromorphic) continuation of zeta functions, we also derive identities and formulas including Bernoulli numbers and Apostol-Bernoulli numbers. Moreover, we give relations between number-theoretic functions and the Fourier expansion of the Eisenstein series. Finally, we give some observations and remarks on these functions.","publication_date":{"day":25,"month":6,"year":2018,"errors":{}}},"translated_abstract":"The aim of this paper is to define some new number-theoretic functions including necklaces polynomials and the numbers of special words such as Lyndon words. By using Dirichlet convolution formula with well-known number-theoretic functions, we derive some new identities and relations associated with Dirichlet series, Lambert series, and also the family of zeta functions including the Riemann zeta functions and polylogarithm functions. By using analytic (meromorphic) continuation of zeta functions, we also derive identities and formulas including Bernoulli numbers and Apostol-Bernoulli numbers. Moreover, we give relations between number-theoretic functions and the Fourier expansion of the Eisenstein series. Finally, we give some observations and remarks on these functions.","internal_url":"https://www.academia.edu/77100434/Identities_and_relations_related_to_the_numbers_of_special_words_derived_from_special_series_with_Dirichlet_convolution","translated_internal_url":"","created_at":"2022-04-20T11:12:46.368-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":146832572,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":84559072,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559072/thumbnails/1.jpg","file_name":"1806.09480v1.pdf","download_url":"https://www.academia.edu/attachments/84559072/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Identities_and_relations_related_to_the.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559072/1806.09480v1-libre.pdf?1650478776=\u0026response-content-disposition=attachment%3B+filename%3DIdentities_and_relations_related_to_the.pdf\u0026Expires=1733183071\u0026Signature=c9tyxvNusjRf8VW0IaYG4p-zDXqO-X7zed5LP3MPy7NlDRU9aPKIYE6tske6AW21lwPpVd4EhDcEgLjZQh1Q~0R6hy~akGX7Argjblqzw4wMr2rlVJSx2Dpyx0-jRyvsjRLZCJI~gRqZVxpLl6eCXiTGaCPW5NS3GRMpmSpZ1Zss0yA~DL~sewfLaURlq11Quz5dPuqVac8qXiE0qnuEZ5HZ7Abd1Ue3YLicbOlpC8wZb2JfA2NMotaCU8Adw9Lbl0HR4QslDsyDWmE7tx5Ff5flSZ7qJIIy7A7fz-XrwQaXE4jUwg4JTs17yWLM0kNdemt9751hn-hYN2-bZgbZkw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Identities_and_relations_related_to_the_numbers_of_special_words_derived_from_special_series_with_Dirichlet_convolution","translated_slug":"","page_count":20,"language":"en","content_type":"Work","owner":{"id":146832572,"first_name":"Yilmaz","middle_initials":null,"last_name":"Simsek","page_name":"YilmazSimsek2","domain_name":"independent","created_at":"2020-02-22T04:58:30.104-08:00","display_name":"Yilmaz Simsek","url":"https://independent.academia.edu/YilmazSimsek2"},"attachments":[{"id":84559072,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559072/thumbnails/1.jpg","file_name":"1806.09480v1.pdf","download_url":"https://www.academia.edu/attachments/84559072/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Identities_and_relations_related_to_the.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559072/1806.09480v1-libre.pdf?1650478776=\u0026response-content-disposition=attachment%3B+filename%3DIdentities_and_relations_related_to_the.pdf\u0026Expires=1733183071\u0026Signature=c9tyxvNusjRf8VW0IaYG4p-zDXqO-X7zed5LP3MPy7NlDRU9aPKIYE6tske6AW21lwPpVd4EhDcEgLjZQh1Q~0R6hy~akGX7Argjblqzw4wMr2rlVJSx2Dpyx0-jRyvsjRLZCJI~gRqZVxpLl6eCXiTGaCPW5NS3GRMpmSpZ1Zss0yA~DL~sewfLaURlq11Quz5dPuqVac8qXiE0qnuEZ5HZ7Abd1Ue3YLicbOlpC8wZb2JfA2NMotaCU8Adw9Lbl0HR4QslDsyDWmE7tx5Ff5flSZ7qJIIy7A7fz-XrwQaXE4jUwg4JTs17yWLM0kNdemt9751hn-hYN2-bZgbZkw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":84559073,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559073/thumbnails/1.jpg","file_name":"1806.09480v1.pdf","download_url":"https://www.academia.edu/attachments/84559073/download_file","bulk_download_file_name":"Identities_and_relations_related_to_the.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559073/1806.09480v1-libre.pdf?1650478777=\u0026response-content-disposition=attachment%3B+filename%3DIdentities_and_relations_related_to_the.pdf\u0026Expires=1733183071\u0026Signature=XjEz6AsJkkmCuNcGgw3wu0Ru5FLNmORnbFkVwi5k9uSBf2~dpRMugCF-Q23zRgbUfVD1NlxRL2LvLHiXZtGM-KkfjPdYRzitaYaytqZzqmk8BQc-QR5WdGSBbXXPuFCQ29Ah7MPZDh2QfZXxgZi-u3SzXVGRgl5OUhWm6KtJMrqaF-wd50JCQNoVsEHvd3NdCInPffxNt~ov9WF4j~nPa8FVEquJ0ixZtGRyk-ILfabNaEW~ICNdbL~FtzQzKwZZnBuwJDtFB4Wopu-jdTOEC013HcTHUzyl8GEepDLhLIjxVxB8lErbqJlivcsGwkjNXe3RIPO0vrC-1ZJwMWfYVw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"}],"urls":[{"id":19721503,"url":"https://arxiv.org/pdf/1806.09480v1.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="77100433"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/77100433/Combinatorial_identities_associated_with_new_families_of_the_numbers_and_polynomials_and_their_approximation_values"><img alt="Research paper thumbnail of Combinatorial identities associated with new families of the numbers and polynomials and their approximation values" class="work-thumbnail" src="https://attachments.academia-assets.com/84559070/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/77100433/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><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Recently, the numbers Y_n(λ ) and the polynomials Y_n(x,λ) have been introduced by the second aut...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Recently, the numbers Y_n(λ ) and the polynomials Y_n(x,λ) have been introduced by the second author [22]. The purpose of this paper is to construct higher-order of these numbers and polynomials with their generating functions. By using these generating functions with their functional equations and derivative equations, we derive various identities and relations including two recurrence relations, Vandermonde type convolution formula, combinatorial sums, the Bernstein basis functions, and also some well known families of special numbers and their interpolation functions such as the Apostol--Bernoulli numbers, the Apostol--Euler numbers, the Stirling numbers of the first kind, and the zeta type function. Finally, by using Stirling&#39;s approximation for factorials, we investigate some approximation values of the special case of the numbers Y_n( λ) .</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="c8001a5ca7c18c80fc7764cc75e966b5" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":84559070,"asset_id":77100433,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/84559070/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77100433"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77100433"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77100433; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77100433]").text(description); $(".js-view-count[data-work-id=77100433]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77100433; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77100433']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77100433, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "c8001a5ca7c18c80fc7764cc75e966b5" } } $('.js-work-strip[data-work-id=77100433]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77100433,"title":"Combinatorial identities associated with new families of the numbers and polynomials and their approximation values","translated_title":"","metadata":{"abstract":"Recently, the numbers Y_n(λ ) and the polynomials Y_n(x,λ) have been introduced by the second author [22]. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="77100431"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/77100431/Generating_functions_for_finite_sums_involving_higher_powers_of_binomial_coefficients_Analysis_of_hypergeometric_functions_including_new_families_of_polynomials_and_numbers"><img alt="Research paper thumbnail of Generating functions for finite sums involving higher powers of binomial coefficients: Analysis of hypergeometric functions including new families of polynomials and numbers" class="work-thumbnail" src="https://attachments.academia-assets.com/84559066/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/77100431/Generating_functions_for_finite_sums_involving_higher_powers_of_binomial_coefficients_Analysis_of_hypergeometric_functions_including_new_families_of_polynomials_and_numbers">Generating functions for finite sums involving higher powers of binomial coefficients: Analysis of hypergeometric functions including new families of polynomials and numbers</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The origin of this study is based on not only explicit formulas of finite sums involving higher p...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The origin of this study is based on not only explicit formulas of finite sums involving higher powers of binomial coefficients, but also explicit evaluations of generating functions for this sums. It should be emphasized that this study contains both new results and literature surveys about some of the related results that have existed so far. With the aid of hypergeometric function, generating functions for a new family of the combinatorial numbers, related to finite sums of powers of binomial coefficients, are constructed. By using these generating functions, a number of new identities have been obtained and at the same time previously well-known formulas and identities have been generalized. Moreover, on this occasion, we identify new families of polynomials including some families of well-known numbers such as Bernoulli numbers, Euler numbers, Stirling numbers, Franel numbers, Catalan numbers, Changhee numbers, Daehee numbers and the others, and also for the polynomials such as...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="3dd2e18c7781a95c56c085886d2269b3" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":84559066,"asset_id":77100431,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/84559066/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77100431"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77100431"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77100431; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77100431]").text(description); $(".js-view-count[data-work-id=77100431]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77100431; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77100431']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77100431, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "3dd2e18c7781a95c56c085886d2269b3" } } $('.js-work-strip[data-work-id=77100431]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77100431,"title":"Generating functions for finite sums involving higher powers of binomial coefficients: Analysis of hypergeometric functions including new families of polynomials and numbers","translated_title":"","metadata":{"abstract":"The origin of this study is based on not only explicit formulas of finite sums involving higher powers of binomial coefficients, but also explicit evaluations of generating functions for this sums. It should be emphasized that this study contains both new results and literature surveys about some of the related results that have existed so far. With the aid of hypergeometric function, generating functions for a new family of the combinatorial numbers, related to finite sums of powers of binomial coefficients, are constructed. By using these generating functions, a number of new identities have been obtained and at the same time previously well-known formulas and identities have been generalized. Moreover, on this occasion, we identify new families of polynomials including some families of well-known numbers such as Bernoulli numbers, Euler numbers, Stirling numbers, Franel numbers, Catalan numbers, Changhee numbers, Daehee numbers and the others, and also for the polynomials such as...","publication_date":{"day":4,"month":1,"year":2019,"errors":{}}},"translated_abstract":"The origin of this study is based on not only explicit formulas of finite sums involving higher powers of binomial coefficients, but also explicit evaluations of generating functions for this sums. It should be emphasized that this study contains both new results and literature surveys about some of the related results that have existed so far. With the aid of hypergeometric function, generating functions for a new family of the combinatorial numbers, related to finite sums of powers of binomial coefficients, are constructed. By using these generating functions, a number of new identities have been obtained and at the same time previously well-known formulas and identities have been generalized. Moreover, on this occasion, we identify new families of polynomials including some families of well-known numbers such as Bernoulli numbers, Euler numbers, Stirling numbers, Franel numbers, Catalan numbers, Changhee numbers, Daehee numbers and the others, and also for the polynomials such as...","internal_url":"https://www.academia.edu/77100431/Generating_functions_for_finite_sums_involving_higher_powers_of_binomial_coefficients_Analysis_of_hypergeometric_functions_including_new_families_of_polynomials_and_numbers","translated_internal_url":"","created_at":"2022-04-20T11:12:45.594-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":146832572,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":84559066,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559066/thumbnails/1.jpg","file_name":"1901.02912v1.pdf","download_url":"https://www.academia.edu/attachments/84559066/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Generating_functions_for_finite_sums_inv.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559066/1901.02912v1-libre.pdf?1650478775=\u0026response-content-disposition=attachment%3B+filename%3DGenerating_functions_for_finite_sums_inv.pdf\u0026Expires=1733183071\u0026Signature=EdD7~Bsh9xsvqNam5c3JPtq4JFPCpHHaPZtupA86JooqoJHrf7fahUHlEjqjnsNEl6PeMtT7roJ0MzPdvtGSCk04BYXy1YpkhlM3uT4yoNvIj~ePSIjdkKall~4WmUw9-op7IboY3xOs5EaN26LzhVuyTWD-McVib2ecrqherV0IMCKw491h6FPVTXyDcqWFIC~gKaUDBeC5x7noAM85s5sNRpSQbsEFLJrQS94mn0TODpEGp~FR2NR4qXBZaX5gaRj~usSEoGbyWbJZAm3iRTvAb-rDjHGKz~gi3U1go5wyGkId9VofzMJUFuDVubCu5GF~Xwj3nOb8E544BmREEA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Generating_functions_for_finite_sums_involving_higher_powers_of_binomial_coefficients_Analysis_of_hypergeometric_functions_including_new_families_of_polynomials_and_numbers","translated_slug":"","page_count":36,"language":"en","content_type":"Work","owner":{"id":146832572,"first_name":"Yilmaz","middle_initials":null,"last_name":"Simsek","page_name":"YilmazSimsek2","domain_name":"independent","created_at":"2020-02-22T04:58:30.104-08:00","display_name":"Yilmaz Simsek","url":"https://independent.academia.edu/YilmazSimsek2"},"attachments":[{"id":84559066,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559066/thumbnails/1.jpg","file_name":"1901.02912v1.pdf","download_url":"https://www.academia.edu/attachments/84559066/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Generating_functions_for_finite_sums_inv.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559066/1901.02912v1-libre.pdf?1650478775=\u0026response-content-disposition=attachment%3B+filename%3DGenerating_functions_for_finite_sums_inv.pdf\u0026Expires=1733183071\u0026Signature=EdD7~Bsh9xsvqNam5c3JPtq4JFPCpHHaPZtupA86JooqoJHrf7fahUHlEjqjnsNEl6PeMtT7roJ0MzPdvtGSCk04BYXy1YpkhlM3uT4yoNvIj~ePSIjdkKall~4WmUw9-op7IboY3xOs5EaN26LzhVuyTWD-McVib2ecrqherV0IMCKw491h6FPVTXyDcqWFIC~gKaUDBeC5x7noAM85s5sNRpSQbsEFLJrQS94mn0TODpEGp~FR2NR4qXBZaX5gaRj~usSEoGbyWbJZAm3iRTvAb-rDjHGKz~gi3U1go5wyGkId9VofzMJUFuDVubCu5GF~Xwj3nOb8E544BmREEA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":84559067,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/84559067/thumbnails/1.jpg","file_name":"1901.02912v1.pdf","download_url":"https://www.academia.edu/attachments/84559067/download_file","bulk_download_file_name":"Generating_functions_for_finite_sums_inv.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/84559067/1901.02912v1-libre.pdf?1650478776=\u0026response-content-disposition=attachment%3B+filename%3DGenerating_functions_for_finite_sums_inv.pdf\u0026Expires=1733183071\u0026Signature=FuIiZx9lRprinbgLbSWFubNlkq7kzJzxxRz2TBjgU7tkJj7tPD1hIOy-BLjFL5gAUu5HaNXwmzynxfBNqnxSfD2RwQN1thoSmvvLuirFlbDphu~kbK9rUOCmlcfeDfgMm-4s1q3oG2ix~BAauGCUM8eo3Dr15Cxs0yM7ykOhEtPJRytw7LSkdHG2fJmk4SdTAy8kL1JXieKKX6Kcnzm-TInhFuiaFcKsdnr43FenQnACN13DcAZpUhQaVIpnOF5jU5tY7uVI-RpaQsRs4ilusmnJYP8BRiObkcXTqb7QqWRdZOZXZsSdsm0dVVI2bdnsZcb0iCyKCSnfZ9O5xWEKDA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":305,"name":"Applied Mathematics","url":"https://www.academia.edu/Documents/in/Applied_Mathematics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":518958,"name":"Mathematical Analysis and Applications","url":"https://www.academia.edu/Documents/in/Mathematical_Analysis_and_Applications"},{"id":1131312,"name":"Academic","url":"https://www.academia.edu/Documents/in/Academic"},{"id":1237788,"name":"Electrical And Electronic Engineering","url":"https://www.academia.edu/Documents/in/Electrical_And_Electronic_Engineering"}],"urls":[{"id":19721500,"url":"https://arxiv.org/pdf/1901.02912v1.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="77100430"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/77100430/Functional_equations_from_generating_functions_a_novel_approach_to_deriving_identities_for_the_Bernstein_basis_functions"><img alt="Research paper thumbnail of Functional equations from generating functions: a novel approach to deriving identities for the Bernstein basis functions" class="work-thumbnail" src="https://attachments.academia-assets.com/84559064/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/77100430/Functional_equations_from_generating_functions_a_novel_approach_to_deriving_identities_for_the_Bernstein_basis_functions">Functional equations from generating functions: a novel approach to deriving identities for the Bernstein basis functions</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The main aim of this paper is to provide a novel approach to deriving identities for the Bernstei...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The main aim of this paper is to provide a novel approach to deriving identities for the Bernstein polynomials using functional equations. We derive various functional equations and differential equations using generating functions. Applying these equations, we give new proofs for some standard identities for the Bernstein basis functions, including formulas for sums, alternating sums, recursion, subdivision, degree raising, differentiation and a formula for the monomials in terms of the Bernstein basis functions. We also derive many new identities for the Bernstein basis functions based on this approach. Moreover, by applying the Laplace transform to the generating functions for the Bernstein basis functions, we obtain some interesting series representations for the Bernstein basis functions.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="17b59aa3102458f819acb536909f66b0" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":84559064,"asset_id":77100430,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/84559064/download_file?st=MTczMzE5MDU4Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77100430"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77100430"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77100430; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77100430]").text(description); $(".js-view-count[data-work-id=77100430]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77100430; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77100430']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77100430, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "17b59aa3102458f819acb536909f66b0" } } $('.js-work-strip[data-work-id=77100430]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77100430,"title":"Functional equations from generating functions: a novel approach to deriving identities for the Bernstein basis functions","translated_title":"","metadata":{"abstract":"The main aim of this paper is to provide a novel approach to deriving identities for the Bernstein polynomials using functional equations. We derive various functional equations and differential equations using generating functions. Applying these equations, we give new proofs for some standard identities for the Bernstein basis functions, including formulas for sums, alternating sums, recursion, subdivision, degree raising, differentiation and a formula for the monomials in terms of the Bernstein basis functions. We also derive many new identities for the Bernstein basis functions based on this approach. Moreover, by applying the Laplace transform to the generating functions for the Bernstein basis functions, we obtain some interesting series representations for the Bernstein basis functions.","publication_date":{"day":21,"month":11,"year":2011,"errors":{}}},"translated_abstract":"The main aim of this paper is to provide a novel approach to deriving identities for the Bernstein polynomials using functional equations. We derive various functional equations and differential equations using generating functions. Applying these equations, we give new proofs for some standard identities for the Bernstein basis functions, including formulas for sums, alternating sums, recursion, subdivision, degree raising, differentiation and a formula for the monomials in terms of the Bernstein basis functions. We also derive many new identities for the Bernstein basis functions based on this approach. 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