{"id":"https://openalex.org/W4388968860","doi":"https://doi.org/10.3390/sym15122113","title":"A Note on Incomplete Fibonacci\u2013Lucas Relations","display_name":"A Note on Incomplete Fibonacci\u2013Lucas Relations","publication_year":2023,"publication_date":"2023-11-24","ids":{"openalex":"https://openalex.org/W4388968860","doi":"https://doi.org/10.3390/sym15122113"},"language":"en","primary_location":{"id":"doi:10.3390/sym15122113","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15122113","pdf_url":"https://www.mdpi.com/2073-8994/15/12/2113/pdf?version=1700839535","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/2073-8994/15/12/2113/pdf?version=1700839535","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5025439589","display_name":"Jingyang Zhong","orcid":"https://orcid.org/0009-0007-1298-9433"},"institutions":[{"id":"https://openalex.org/I80947539","display_name":"Fuzhou University","ror":"https://ror.org/011xvna82","country_code":"CN","type":"education","lineage":["https://openalex.org/I80947539"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Jingyang Zhong","raw_affiliation_strings":["School of Mathematics and Statistics, Fuzhou University, Fuzhou 350108, China"],"raw_orcid":"https://orcid.org/0009-0007-1298-9433","affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Fuzhou University, Fuzhou 350108, China","institution_ids":["https://openalex.org/I80947539"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5089382976","display_name":"Jialing Yao","orcid":null},"institutions":[{"id":"https://openalex.org/I80947539","display_name":"Fuzhou University","ror":"https://ror.org/011xvna82","country_code":"CN","type":"education","lineage":["https://openalex.org/I80947539"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Jialing Yao","raw_affiliation_strings":["School of Mathematics and Statistics, Fuzhou University, Fuzhou 350108, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Fuzhou University, Fuzhou 350108, China","institution_ids":["https://openalex.org/I80947539"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5045680628","display_name":"Chan-Liang Chung","orcid":"https://orcid.org/0000-0001-7548-3607"},"institutions":[{"id":"https://openalex.org/I80947539","display_name":"Fuzhou University","ror":"https://ror.org/011xvna82","country_code":"CN","type":"education","lineage":["https://openalex.org/I80947539"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Chan-Liang Chung","raw_affiliation_strings":["School of Mathematics and Statistics, Fuzhou University, Fuzhou 350108, China"],"raw_orcid":"https://orcid.org/0000-0001-7548-3607","affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Fuzhou University, Fuzhou 350108, China","institution_ids":["https://openalex.org/I80947539"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5045680628"],"corresponding_institution_ids":["https://openalex.org/I80947539"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2227},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2227},"fwci":0.3779,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.54819171,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":97},"biblio":{"volume":"15","issue":"12","first_page":"2113","last_page":"2113"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12516","display_name":"Advanced Mathematical Theories and Applications","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12516","display_name":"Advanced Mathematical Theories and Applications","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9803000092506409,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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":0.9761000275611877,"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/fibonacci-number","display_name":"Fibonacci number","score":0.9046680927276611},{"id":"https://openalex.org/keywords/fibonacci-polynomials","display_name":"Fibonacci polynomials","score":0.9033621549606323},{"id":"https://openalex.org/keywords/lucas-number","display_name":"Lucas number","score":0.8287979960441589},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6758264303207397},{"id":"https://openalex.org/keywords/relation","display_name":"Relation (database)","score":0.5988346338272095},{"id":"https://openalex.org/keywords/bivariate-analysis","display_name":"Bivariate analysis","score":0.5947287082672119},{"id":"https://openalex.org/keywords/lucas-sequence","display_name":"Lucas sequence","score":0.549207866191864},{"id":"https://openalex.org/keywords/recurrence-relation","display_name":"Recurrence relation","score":0.5107138156890869},{"id":"https://openalex.org/keywords/pisano-period","display_name":"Pisano period","score":0.4230201244354248},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4053250551223755},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.36234891414642334},{"id":"https://openalex.org/keywords/orthogonal-polynomials","display_name":"Orthogonal polynomials","score":0.13899850845336914},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.13780441880226135},{"id":"https://openalex.org/keywords/classical-orthogonal-polynomials","display_name":"Classical orthogonal polynomials","score":0.13767153024673462},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.08176758885383606}],"concepts":[{"id":"https://openalex.org/C173734053","wikidata":"https://www.wikidata.org/wiki/Q47577","display_name":"Fibonacci number","level":2,"score":0.9046680927276611},{"id":"https://openalex.org/C79117305","wikidata":"https://www.wikidata.org/wiki/Q974295","display_name":"Fibonacci polynomials","level":4,"score":0.9033621549606323},{"id":"https://openalex.org/C3634175","wikidata":"https://www.wikidata.org/wiki/Q2503280","display_name":"Lucas number","level":3,"score":0.8287979960441589},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6758264303207397},{"id":"https://openalex.org/C25343380","wikidata":"https://www.wikidata.org/wiki/Q277521","display_name":"Relation (database)","level":2,"score":0.5988346338272095},{"id":"https://openalex.org/C64341305","wikidata":"https://www.wikidata.org/wiki/Q4919225","display_name":"Bivariate analysis","level":2,"score":0.5947287082672119},{"id":"https://openalex.org/C154977054","wikidata":"https://www.wikidata.org/wiki/Q1759646","display_name":"Lucas sequence","level":5,"score":0.549207866191864},{"id":"https://openalex.org/C201292218","wikidata":"https://www.wikidata.org/wiki/Q740970","display_name":"Recurrence relation","level":2,"score":0.5107138156890869},{"id":"https://openalex.org/C174757984","wikidata":"https://www.wikidata.org/wiki/Q4351325","display_name":"Pisano period","level":5,"score":0.4230201244354248},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4053250551223755},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.36234891414642334},{"id":"https://openalex.org/C10628310","wikidata":"https://www.wikidata.org/wiki/Q619458","display_name":"Orthogonal polynomials","level":2,"score":0.13899850845336914},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.13780441880226135},{"id":"https://openalex.org/C21736991","wikidata":"https://www.wikidata.org/wiki/Q17006917","display_name":"Classical orthogonal polynomials","level":3,"score":0.13767153024673462},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.08176758885383606},{"id":"https://openalex.org/C77088390","wikidata":"https://www.wikidata.org/wiki/Q8513","display_name":"Database","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/sym15122113","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15122113","pdf_url":"https://www.mdpi.com/2073-8994/15/12/2113/pdf?version=1700839535","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:727867acc45542d0b1e03d0abe5ba6b6","is_oa":false,"landing_page_url":"https://doaj.org/article/727867acc45542d0b1e03d0abe5ba6b6","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 15, Iss 12, p 2113 (2023)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/sym15122113","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15122113","pdf_url":"https://www.mdpi.com/2073-8994/15/12/2113/pdf?version=1700839535","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":[{"id":"https://openalex.org/G2579799103","display_name":null,"funder_award_id":"XRC-20077:0030-50009112","funder_id":"https://openalex.org/F4320325408","funder_display_name":"Fuzhou University"}],"funders":[{"id":"https://openalex.org/F4320325408","display_name":"Fuzhou University","ror":"https://ror.org/011xvna82"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4388968860.pdf","grobid_xml":"https://content.openalex.org/works/W4388968860.grobid-xml"},"referenced_works_count":8,"referenced_works":["https://openalex.org/W1983737942","https://openalex.org/W2013273021","https://openalex.org/W2018632826","https://openalex.org/W2396203592","https://openalex.org/W2967611107","https://openalex.org/W4379091999","https://openalex.org/W6848476052","https://openalex.org/W6853674616"],"related_works":["https://openalex.org/W3131644302","https://openalex.org/W2350136088","https://openalex.org/W4323048372","https://openalex.org/W2015523789","https://openalex.org/W4205905008","https://openalex.org/W2437285847","https://openalex.org/W2919811827","https://openalex.org/W2591594467","https://openalex.org/W1983874533","https://openalex.org/W3173275317"],"abstract_inverted_index":{"We":[0,15,69],"define":[1],"the":[2,9,36,45,60,63,74,78],"incomplete":[3,10,33,48,64,75,79],"generalized":[4,11,49],"bivariate":[5,12,50],"Fibonacci":[6,51],"p-polynomials":[7],"and":[8,20,40,52,77],"Lucas":[13,53],"p-polynomials.":[14,54],"study":[16],"their":[17,26],"recursive":[18],"relations":[19],"derive":[21],"an":[22,32,83],"interesting":[23],"relationship":[24],"through":[25],"generating":[27],"functions.":[28],"Subsequently,":[29],"we":[30],"prove":[31],"version":[34],"of":[35],"well-known":[37],"Fibonacci\u2013Lucas":[38,65],"relation":[39,46,66,72],"make":[41],"some":[42],"extensions":[43],"to":[44,62],"involving":[47,73],"An":[55],"argument":[56],"about":[57],"going":[58],"from":[59],"regular":[61],"is":[67],"discussed.":[68],"provide":[70],"a":[71],"Leonardo":[76],"Lucas\u2013Leonardo":[80],"p-numbers":[81],"as":[82],"illustration.":[84]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2024,"cited_by_count":2}],"updated_date":"2026-08-18T07:49:30.821534","created_date":"2025-10-10T00:00:00"}
