{"id":"https://openalex.org/W2035458920","doi":"https://doi.org/10.1090/s0025-5718-09-02230-3","title":"Fourier expansions and integral representations for the Apostol-Bernoulli and Apostol-Euler polynomials","display_name":"Fourier expansions and integral representations for the Apostol-Bernoulli and Apostol-Euler polynomials","publication_year":2009,"publication_date":"2009-06-12","ids":{"openalex":"https://openalex.org/W2035458920","doi":"https://doi.org/10.1090/s0025-5718-09-02230-3","mag":"2035458920"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-09-02230-3","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-09-02230-3","pdf_url":"https://www.ams.org/mcom/2009-78-268/S0025-5718-09-02230-3/S0025-5718-09-02230-3.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://www.ams.org/mcom/2009-78-268/S0025-5718-09-02230-3/S0025-5718-09-02230-3.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5073211907","display_name":"Qiu-Ming Luo","orcid":"https://orcid.org/0000-0002-9248-3884"},"institutions":[{"id":"https://openalex.org/I66867065","display_name":"East China Normal University","ror":"https://ror.org/02n96ep67","country_code":"CN","type":"education","lineage":["https://openalex.org/I66867065"]},{"id":"https://openalex.org/I4210110925","display_name":"Jiaozuo University","ror":"https://ror.org/024nbxn35","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210110925"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Qiu-Ming Luo","raw_affiliation_strings":["Department of Mathematics, East China Normal University, Shanghai 200241, People\u00e2\u0080\u0099s Republic of China \u00e2\u0080\u0093and\u00e2\u0080\u0093 Department of Mathematics, Jiaozuo University, Henan Jiaozuo 454003, People\u00e2\u0080\u0099s Republic of China","Department of Mathematics, East China Normal University, Shanghai 200241, People\u2019s Republic of China \u2013and\u2013 Department of Mathematics, Jiaozuo University, Henan Jiaozuo 454003, People\u2019s Republic of China"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, East China Normal University, Shanghai 200241, People\u00e2\u0080\u0099s Republic of China \u00e2\u0080\u0093and\u00e2\u0080\u0093 Department of Mathematics, Jiaozuo University, Henan Jiaozuo 454003, People\u00e2\u0080\u0099s Republic of China","institution_ids":["https://openalex.org/I4210110925","https://openalex.org/I66867065"]},{"raw_affiliation_string":"Department of Mathematics, East China Normal University, Shanghai 200241, People\u2019s Republic of China \u2013and\u2013 Department of Mathematics, Jiaozuo University, Henan Jiaozuo 454003, People\u2019s Republic of China","institution_ids":["https://openalex.org/I66867065"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5073211907"],"corresponding_institution_ids":["https://openalex.org/I4210110925","https://openalex.org/I66867065"],"apc_list":null,"apc_paid":null,"fwci":9.9797,"has_fulltext":true,"cited_by_count":71,"citation_normalized_percentile":{"value":0.99007055,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":91,"max":99},"biblio":{"volume":"78","issue":"268","first_page":"2193","last_page":"2208"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11428","display_name":"Advanced Mathematical Identities","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11428","display_name":"Advanced Mathematical Identities","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11191","display_name":"Mathematical functions and polynomials","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.9987000226974487,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8979161977767944},{"id":"https://openalex.org/keywords/bernoulli-polynomials","display_name":"Bernoulli polynomials","score":0.7109937071800232},{"id":"https://openalex.org/keywords/discrete-orthogonal-polynomials","display_name":"Discrete orthogonal polynomials","score":0.5560088157653809},{"id":"https://openalex.org/keywords/classical-orthogonal-polynomials","display_name":"Classical orthogonal polynomials","score":0.5523831248283386},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.5191113948822021},{"id":"https://openalex.org/keywords/orthogonal-polynomials","display_name":"Orthogonal polynomials","score":0.4830537438392639},{"id":"https://openalex.org/keywords/wilson-polynomials","display_name":"Wilson polynomials","score":0.4718458652496338},{"id":"https://openalex.org/keywords/difference-polynomials","display_name":"Difference polynomials","score":0.4359388053417206},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.36202484369277954}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8979161977767944},{"id":"https://openalex.org/C196172290","wikidata":"https://www.wikidata.org/wiki/Q2346201","display_name":"Bernoulli polynomials","level":4,"score":0.7109937071800232},{"id":"https://openalex.org/C52537462","wikidata":"https://www.wikidata.org/wiki/Q5282055","display_name":"Discrete orthogonal polynomials","level":3,"score":0.5560088157653809},{"id":"https://openalex.org/C21736991","wikidata":"https://www.wikidata.org/wiki/Q17006917","display_name":"Classical orthogonal polynomials","level":3,"score":0.5523831248283386},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5191113948822021},{"id":"https://openalex.org/C10628310","wikidata":"https://www.wikidata.org/wiki/Q619458","display_name":"Orthogonal polynomials","level":2,"score":0.4830537438392639},{"id":"https://openalex.org/C43176163","wikidata":"https://www.wikidata.org/wiki/Q8023263","display_name":"Wilson polynomials","level":4,"score":0.4718458652496338},{"id":"https://openalex.org/C86607863","wikidata":"https://www.wikidata.org/wiki/Q5275279","display_name":"Difference polynomials","level":3,"score":0.4359388053417206},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.36202484369277954}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1090/s0025-5718-09-02230-3","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-09-02230-3","pdf_url":"https://www.ams.org/mcom/2009-78-268/S0025-5718-09-02230-3/S0025-5718-09-02230-3.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.160.4523","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.160.4523","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.ams.org/mcom/2009-78-268/S0025-5718-09-02230-3/S0025-5718-09-02230-3.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-09-02230-3","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-09-02230-3","pdf_url":"https://www.ams.org/mcom/2009-78-268/S0025-5718-09-02230-3/S0025-5718-09-02230-3.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3700339054","display_name":null,"funder_award_id":"PCSIRT","funder_id":"https://openalex.org/F4320321106","funder_display_name":"Ministry of Education of the People's Republic of China"}],"funders":[{"id":"https://openalex.org/F4320321106","display_name":"Ministry of Education of the People's Republic of China","ror":"https://ror.org/01mv9t934"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2035458920.pdf","grobid_xml":"https://content.openalex.org/works/W2035458920.grobid-xml"},"referenced_works_count":24,"referenced_works":["https://openalex.org/W30589444","https://openalex.org/W205771879","https://openalex.org/W1488862788","https://openalex.org/W1497283712","https://openalex.org/W1556192255","https://openalex.org/W1647973296","https://openalex.org/W1865602457","https://openalex.org/W1984670355","https://openalex.org/W1985053528","https://openalex.org/W2008439130","https://openalex.org/W2013907652","https://openalex.org/W2018273723","https://openalex.org/W2021111383","https://openalex.org/W2022582059","https://openalex.org/W2044270276","https://openalex.org/W2045698952","https://openalex.org/W2066875849","https://openalex.org/W2073734991","https://openalex.org/W2083245861","https://openalex.org/W2129840940","https://openalex.org/W2801179766","https://openalex.org/W3127380004","https://openalex.org/W3150402801","https://openalex.org/W4251960606"],"related_works":["https://openalex.org/W2072647369","https://openalex.org/W2168893271","https://openalex.org/W2070537393","https://openalex.org/W4380152001","https://openalex.org/W2045154321","https://openalex.org/W4322760296","https://openalex.org/W1549388788","https://openalex.org/W4298153664","https://openalex.org/W4293785478","https://openalex.org/W4297323781"],"abstract_inverted_index":{"We":[0,20,38],"investigate":[1],"Fourier":[2],"expansions":[3],"for":[4,28,44],"the":[5,11,34,41,45],"Apostol-Bernoulli":[6],"and":[7,15,48,51],"Apostol-Euler":[8],"polynomials":[9,30,50],"using":[10],"Lipschitz":[12],"summation":[13],"formula":[14],"obtain":[16],"their":[17],"integral":[18,42],"representations.":[19],"give":[21],"some":[22],"explicit":[23],"formulas":[24],"at":[25],"rational":[26],"arguments":[27],"these":[29],"in":[31],"terms":[32],"of":[33],"Hurwitz":[35],"zeta":[36],"function.":[37],"also":[39],"derive":[40],"representations":[43],"classical":[46],"Bernoulli":[47],"Euler":[49],"related":[52],"known":[53],"results.":[54]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":4},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":9},{"year":2014,"cited_by_count":5},{"year":2013,"cited_by_count":8},{"year":2012,"cited_by_count":9}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}