{"id":"https://openalex.org/W2007342253","doi":"https://doi.org/10.1016/s0196-8858(02)00020-9","title":"Explicit evaluations and reciprocity theorems for finite trigonometric sums","display_name":"Explicit evaluations and reciprocity theorems for finite trigonometric sums","publication_year":2002,"publication_date":"2002-10-01","ids":{"openalex":"https://openalex.org/W2007342253","doi":"https://doi.org/10.1016/s0196-8858(02)00020-9","mag":"2007342253"},"language":"en","primary_location":{"id":"doi:10.1016/s0196-8858(02)00020-9","is_oa":false,"landing_page_url":"https://doi.org/10.1016/s0196-8858(02)00020-9","pdf_url":null,"source":{"id":"https://openalex.org/S164906323","display_name":"Advances in Applied Mathematics","issn_l":"0196-8858","issn":["0196-8858","1090-2074"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Advances in Applied Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5078409229","display_name":"Bruce C. Berndt","orcid":null},"institutions":[{"id":"https://openalex.org/I157725225","display_name":"University of Illinois Urbana-Champaign","ror":"https://ror.org/047426m28","country_code":"US","type":"education","lineage":["https://openalex.org/I157725225"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Bruce C. Berndt","raw_affiliation_strings":["Department of Mathematics, University of Illinois, 1409 West Green St., Urbana, IL 61801, USA","Department of Mathematics, University of Illinois, 1409 West Green St., Urbana, IL 61801, USA#TAB#"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Illinois, 1409 West Green St., Urbana, IL 61801, USA","institution_ids":["https://openalex.org/I157725225"]},{"raw_affiliation_string":"Department of Mathematics, University of Illinois, 1409 West Green St., Urbana, IL 61801, USA#TAB#","institution_ids":["https://openalex.org/I157725225"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5020411231","display_name":"Boon Pin Yeap","orcid":null},"institutions":[{"id":"https://openalex.org/I157725225","display_name":"University of Illinois Urbana-Champaign","ror":"https://ror.org/047426m28","country_code":"US","type":"education","lineage":["https://openalex.org/I157725225"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Boon Pin Yeap","raw_affiliation_strings":["Department of Mathematics, University of Illinois, 1409 West Green St., Urbana, IL 61801, USA","Department of Mathematics, University of Illinois, 1409 West Green St., Urbana, IL 61801, USA#TAB#"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Illinois, 1409 West Green St., Urbana, IL 61801, USA","institution_ids":["https://openalex.org/I157725225"]},{"raw_affiliation_string":"Department of Mathematics, University of Illinois, 1409 West Green St., Urbana, IL 61801, USA#TAB#","institution_ids":["https://openalex.org/I157725225"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5078409229"],"corresponding_institution_ids":["https://openalex.org/I157725225"],"apc_list":{"value":2670,"currency":"USD","value_usd":2670},"apc_paid":null,"fwci":4.8286,"has_fulltext":false,"cited_by_count":98,"citation_normalized_percentile":{"value":0.94302627,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":100},"biblio":{"volume":"29","issue":"3","first_page":"358","last_page":"385"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.9984999895095825,"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/T11166","display_name":"Analytic Number Theory Research","score":0.9984999895095825,"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/T11428","display_name":"Advanced Mathematical Identities","score":0.9983000159263611,"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":0.9976000189781189,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.9247971773147583},{"id":"https://openalex.org/keywords/trigonometry","display_name":"Trigonometry","score":0.6436751484870911},{"id":"https://openalex.org/keywords/trigonometric-substitution","display_name":"Trigonometric substitution","score":0.5228660106658936},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.4953967332839966},{"id":"https://openalex.org/keywords/reciprocity","display_name":"Reciprocity (cultural anthropology)","score":0.4634881317615509},{"id":"https://openalex.org/keywords/mathematical-proof","display_name":"Mathematical proof","score":0.45750096440315247},{"id":"https://openalex.org/keywords/stern","display_name":"Stern","score":0.4512448012828827},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.43959230184555054},{"id":"https://openalex.org/keywords/trigonometric-series","display_name":"Trigonometric series","score":0.41051995754241943},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.386963427066803},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.33995622396469116},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.3396068811416626},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.32007327675819397},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.14512255787849426}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.9247971773147583},{"id":"https://openalex.org/C29001434","wikidata":"https://www.wikidata.org/wiki/Q8084","display_name":"Trigonometry","level":2,"score":0.6436751484870911},{"id":"https://openalex.org/C9317044","wikidata":"https://www.wikidata.org/wiki/Q1191963","display_name":"Trigonometric substitution","level":5,"score":0.5228660106658936},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.4953967332839966},{"id":"https://openalex.org/C169903001","wikidata":"https://www.wikidata.org/wiki/Q3264987","display_name":"Reciprocity (cultural anthropology)","level":2,"score":0.4634881317615509},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.45750096440315247},{"id":"https://openalex.org/C96597354","wikidata":"https://www.wikidata.org/wiki/Q273062","display_name":"Stern","level":2,"score":0.4512448012828827},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.43959230184555054},{"id":"https://openalex.org/C120362076","wikidata":"https://www.wikidata.org/wiki/Q1149336","display_name":"Trigonometric series","level":2,"score":0.41051995754241943},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.386963427066803},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.33995622396469116},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.3396068811416626},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.32007327675819397},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.14512255787849426},{"id":"https://openalex.org/C171836373","wikidata":"https://www.wikidata.org/wiki/Q2266329","display_name":"Linear interpolation","level":3,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C15744967","wikidata":"https://www.wikidata.org/wiki/Q9418","display_name":"Psychology","level":0,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.0},{"id":"https://openalex.org/C77805123","wikidata":"https://www.wikidata.org/wiki/Q161272","display_name":"Social psychology","level":1,"score":0.0},{"id":"https://openalex.org/C49608258","wikidata":"https://www.wikidata.org/wiki/Q611705","display_name":"Bicubic interpolation","level":4,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C199104240","wikidata":"https://www.wikidata.org/wiki/Q118291","display_name":"Marine engineering","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1016/s0196-8858(02)00020-9","is_oa":false,"landing_page_url":"https://doi.org/10.1016/s0196-8858(02)00020-9","pdf_url":null,"source":{"id":"https://openalex.org/S164906323","display_name":"Advances in Applied Mathematics","issn_l":"0196-8858","issn":["0196-8858","1090-2074"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Advances in Applied Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.92.5542","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.92.5542","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.uiuc.edu/~berndt/articles/trigsums.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.5699999928474426,"display_name":"Peace, Justice and strong institutions","id":"https://metadata.un.org/sdg/16"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":89,"referenced_works":["https://openalex.org/W40488325","https://openalex.org/W148540366","https://openalex.org/W929637328","https://openalex.org/W1481596509","https://openalex.org/W1516189574","https://openalex.org/W1538404615","https://openalex.org/W1543216622","https://openalex.org/W1578953313","https://openalex.org/W1580088620","https://openalex.org/W1731159951","https://openalex.org/W1964464432","https://openalex.org/W1967915596","https://openalex.org/W1971298600","https://openalex.org/W1972786864","https://openalex.org/W1974945857","https://openalex.org/W1978003832","https://openalex.org/W1979818457","https://openalex.org/W1984523785","https://openalex.org/W1994068306","https://openalex.org/W1997838401","https://openalex.org/W2004072201","https://openalex.org/W2010163332","https://openalex.org/W2010646630","https://openalex.org/W2011195691","https://openalex.org/W2013815868","https://openalex.org/W2014778061","https://openalex.org/W2015898674","https://openalex.org/W2026870825","https://openalex.org/W2026936067","https://openalex.org/W2027274761","https://openalex.org/W2030223747","https://openalex.org/W2030458667","https://openalex.org/W2033289104","https://openalex.org/W2040371959","https://openalex.org/W2051353259","https://openalex.org/W2054673549","https://openalex.org/W2056508498","https://openalex.org/W2057182494","https://openalex.org/W2059647931","https://openalex.org/W2060202753","https://openalex.org/W2060994055","https://openalex.org/W2061962785","https://openalex.org/W2066033242","https://openalex.org/W2070286322","https://openalex.org/W2076597612","https://openalex.org/W2077827904","https://openalex.org/W2080464046","https://openalex.org/W2083412983","https://openalex.org/W2091929090","https://openalex.org/W2093770144","https://openalex.org/W2125514810","https://openalex.org/W2129191013","https://openalex.org/W2137371238","https://openalex.org/W2146116040","https://openalex.org/W2148712836","https://openalex.org/W2158048440","https://openalex.org/W2170576511","https://openalex.org/W2312233552","https://openalex.org/W2319682107","https://openalex.org/W2322527513","https://openalex.org/W2327234026","https://openalex.org/W2328069953","https://openalex.org/W2334305095","https://openalex.org/W2393675882","https://openalex.org/W2501864437","https://openalex.org/W2788012517","https://openalex.org/W2796825073","https://openalex.org/W2949784519","https://openalex.org/W2951757019","https://openalex.org/W2964327996","https://openalex.org/W2994012076","https://openalex.org/W3021227759","https://openalex.org/W3105213090","https://openalex.org/W3140509714","https://openalex.org/W3175367423","https://openalex.org/W4229836012","https://openalex.org/W4233046626","https://openalex.org/W4233325724","https://openalex.org/W4233862013","https://openalex.org/W4240656383","https://openalex.org/W4244040791","https://openalex.org/W4251359466","https://openalex.org/W4253771001","https://openalex.org/W4256049187","https://openalex.org/W4285719527","https://openalex.org/W4299396019","https://openalex.org/W6601680369","https://openalex.org/W6724472164","https://openalex.org/W6771792572"],"related_works":["https://openalex.org/W164200052","https://openalex.org/W1483355072","https://openalex.org/W2898978554","https://openalex.org/W4289328889","https://openalex.org/W3022244109","https://openalex.org/W579400037","https://openalex.org/W3104094566","https://openalex.org/W2981512670","https://openalex.org/W3022590230","https://openalex.org/W4387010780"],"abstract_inverted_index":null,"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":7},{"year":2023,"cited_by_count":8},{"year":2022,"cited_by_count":7},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":7},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":5},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":15},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":3}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}