{"id":"https://openalex.org/W2905260770","doi":"https://doi.org/10.3390/sym10120742","title":"Representation by Chebyshev Polynomials for Sums of Finite Products of Chebyshev Polynomials","display_name":"Representation by Chebyshev Polynomials for Sums of Finite Products of Chebyshev Polynomials","publication_year":2018,"publication_date":"2018-12-11","ids":{"openalex":"https://openalex.org/W2905260770","doi":"https://doi.org/10.3390/sym10120742","mag":"2905260770"},"language":"en","primary_location":{"id":"doi:10.3390/sym10120742","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym10120742","pdf_url":"https://www.mdpi.com/2073-8994/10/12/742/pdf?version=1545286873","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/10/12/742/pdf?version=1545286873","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100653780","display_name":"Taekyun Kim","orcid":"https://orcid.org/0000-0002-3731-7023"},"institutions":[{"id":"https://openalex.org/I161024014","display_name":"Kwangwoon University","ror":"https://ror.org/02e9zc863","country_code":"KR","type":"education","lineage":["https://openalex.org/I161024014"]}],"countries":["KR"],"is_corresponding":false,"raw_author_name":"Taekyun Kim","raw_affiliation_strings":["Department of Mathematics, Kwangwoon University, Seoul 139-701, Korea"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Kwangwoon University, Seoul 139-701, Korea","institution_ids":["https://openalex.org/I161024014"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5003571699","display_name":"Dae San Kim","orcid":"https://orcid.org/0000-0001-9599-7015"},"institutions":[{"id":"https://openalex.org/I148751991","display_name":"Sogang University","ror":"https://ror.org/056tn4839","country_code":"KR","type":"education","lineage":["https://openalex.org/I148751991"]}],"countries":["KR"],"is_corresponding":false,"raw_author_name":"Dae San Kim","raw_affiliation_strings":["Department of Mathematics, Sogang University, Seoul 121-742, Korea"],"raw_orcid":"https://orcid.org/0000-0001-9599-7015","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Sogang University, Seoul 121-742, Korea","institution_ids":["https://openalex.org/I148751991"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5049662161","display_name":"Lee-Chae Jang","orcid":"https://orcid.org/0000-0001-5317-1725"},"institutions":[{"id":"https://openalex.org/I24062138","display_name":"Konkuk University","ror":"https://ror.org/025h1m602","country_code":"KR","type":"education","lineage":["https://openalex.org/I24062138"]}],"countries":["KR"],"is_corresponding":true,"raw_author_name":"Lee-Chae Jang","raw_affiliation_strings":["Graduate School of Education, Konkuk University, Seoul 139-701, Korea"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Graduate School of Education, Konkuk University, Seoul 139-701, Korea","institution_ids":["https://openalex.org/I24062138"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5113753827","display_name":"Dmitry V. Dolgy","orcid":null},"institutions":[{"id":"https://openalex.org/I161024014","display_name":"Kwangwoon University","ror":"https://ror.org/02e9zc863","country_code":"KR","type":"education","lineage":["https://openalex.org/I161024014"]}],"countries":["KR"],"is_corresponding":false,"raw_author_name":"Dmitry V. Dolgy","raw_affiliation_strings":["Hanrimwon, Kwangwoon University, Seoul 139-701, Korea"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Hanrimwon, Kwangwoon University, Seoul 139-701, Korea","institution_ids":["https://openalex.org/I161024014"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5049662161"],"corresponding_institution_ids":["https://openalex.org/I24062138"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":2.0772,"has_fulltext":false,"cited_by_count":7,"citation_normalized_percentile":{"value":0.86898945,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"10","issue":"12","first_page":"742","last_page":"742"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"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"}},"topics":[{"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/T12661","display_name":"Iterative Methods for Nonlinear Equations","score":0.9940999746322632,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9876000285148621,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"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/chebyshev-polynomials","display_name":"Chebyshev polynomials","score":0.7846688032150269},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7776527404785156},{"id":"https://openalex.org/keywords/classical-orthogonal-polynomials","display_name":"Classical orthogonal polynomials","score":0.6522005200386047},{"id":"https://openalex.org/keywords/orthogonal-polynomials","display_name":"Orthogonal polynomials","score":0.5577203631401062},{"id":"https://openalex.org/keywords/discrete-orthogonal-polynomials","display_name":"Discrete orthogonal polynomials","score":0.5254637002944946},{"id":"https://openalex.org/keywords/gegenbauer-polynomials","display_name":"Gegenbauer polynomials","score":0.5147709846496582},{"id":"https://openalex.org/keywords/jacobi-polynomials","display_name":"Jacobi polynomials","score":0.4725871980190277},{"id":"https://openalex.org/keywords/hahn-polynomials","display_name":"Hahn polynomials","score":0.4714321196079254},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.42663323879241943},{"id":"https://openalex.org/keywords/difference-polynomials","display_name":"Difference polynomials","score":0.42594051361083984},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3509538769721985},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.19682905077934265}],"concepts":[{"id":"https://openalex.org/C129785596","wikidata":"https://www.wikidata.org/wiki/Q619511","display_name":"Chebyshev polynomials","level":2,"score":0.7846688032150269},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7776527404785156},{"id":"https://openalex.org/C21736991","wikidata":"https://www.wikidata.org/wiki/Q17006917","display_name":"Classical orthogonal polynomials","level":3,"score":0.6522005200386047},{"id":"https://openalex.org/C10628310","wikidata":"https://www.wikidata.org/wiki/Q619458","display_name":"Orthogonal polynomials","level":2,"score":0.5577203631401062},{"id":"https://openalex.org/C52537462","wikidata":"https://www.wikidata.org/wiki/Q5282055","display_name":"Discrete orthogonal polynomials","level":3,"score":0.5254637002944946},{"id":"https://openalex.org/C78540521","wikidata":"https://www.wikidata.org/wiki/Q1498246","display_name":"Gegenbauer polynomials","level":4,"score":0.5147709846496582},{"id":"https://openalex.org/C48490523","wikidata":"https://www.wikidata.org/wiki/Q371631","display_name":"Jacobi polynomials","level":3,"score":0.4725871980190277},{"id":"https://openalex.org/C61349858","wikidata":"https://www.wikidata.org/wiki/Q1568814","display_name":"Hahn polynomials","level":5,"score":0.4714321196079254},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.42663323879241943},{"id":"https://openalex.org/C86607863","wikidata":"https://www.wikidata.org/wiki/Q5275279","display_name":"Difference polynomials","level":3,"score":0.42594051361083984},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3509538769721985},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.19682905077934265}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/sym10120742","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym10120742","pdf_url":"https://www.mdpi.com/2073-8994/10/12/742/pdf?version=1545286873","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:8e4ec0cc183f48068995238188514f22","is_oa":true,"landing_page_url":"https://doaj.org/article/8e4ec0cc183f48068995238188514f22","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 10, Iss 12, p 742 (2018)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/sym10120742","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym10120742","pdf_url":"https://www.mdpi.com/2073-8994/10/12/742/pdf?version=1545286873","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":22,"referenced_works":["https://openalex.org/W1607610601","https://openalex.org/W1990348053","https://openalex.org/W2009936345","https://openalex.org/W2040239232","https://openalex.org/W2156593399","https://openalex.org/W2319147158","https://openalex.org/W2469638860","https://openalex.org/W2510276550","https://openalex.org/W2748928800","https://openalex.org/W2752893298","https://openalex.org/W2754173383","https://openalex.org/W2790229558","https://openalex.org/W2806720594","https://openalex.org/W2810917149","https://openalex.org/W2886616069","https://openalex.org/W2886775003","https://openalex.org/W2901866597","https://openalex.org/W3125267054","https://openalex.org/W3125516916","https://openalex.org/W4254246953","https://openalex.org/W4402831449","https://openalex.org/W6872650189"],"related_works":["https://openalex.org/W2072647369","https://openalex.org/W2168893271","https://openalex.org/W2045154321","https://openalex.org/W2070537393","https://openalex.org/W2132314122","https://openalex.org/W4394818655","https://openalex.org/W2158826266","https://openalex.org/W2052308123","https://openalex.org/W4287241472","https://openalex.org/W2504939013"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3],"consider":[4],"sums":[5],"of":[6,9,12,29,34,37,59,73,76,81,85],"finite":[7],"products":[8],"Chebyshev":[10,35],"polynomials":[11,36,78],"the":[13,23,60,68,71,74],"first,":[14],"third,":[15],"and":[16],"fourth":[17],"kinds,":[18],"which":[19,63],"are":[20,89],"different":[21],"from":[22],"previously-studied":[24],"ones.":[25],"We":[26],"represent":[27],"each":[28],"them":[30],"as":[31,56],"linear":[32],"combinations":[33],"all":[38],"kinds":[39],"whose":[40],"coefficients":[41,69],"involve":[42],"some":[43],"terminating":[44],"hypergeometric":[45],"functions":[46],"2":[47],"F":[48],"1":[49],".":[50],"The":[51],"results":[52],"may":[53],"be":[54],"viewed":[55],"a":[57],"generalization":[58],"linearization":[61],"problem,":[62],"is":[64],"concerned":[65],"with":[66],"determining":[67],"in":[70,79],"expansion":[72],"product":[75],"two":[77],"terms":[80],"any":[82],"given":[83],"sequence":[84],"polynomials.":[86],"These":[87],"representations":[88],"obtained":[90],"by":[91],"explicit":[92],"computations.":[93]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":3}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}