{"id":"https://openalex.org/W4415987086","doi":"https://doi.org/10.3390/sym17111898","title":"Generalized Fibonacci Polynomials and Their Properties","display_name":"Generalized Fibonacci Polynomials and Their Properties","publication_year":2025,"publication_date":"2025-11-07","ids":{"openalex":"https://openalex.org/W4415987086","doi":"https://doi.org/10.3390/sym17111898"},"language":"en","primary_location":{"id":"doi:10.3390/sym17111898","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17111898","pdf_url":"https://www.mdpi.com/2073-8994/17/11/1898/pdf","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"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/17/11/1898/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5050170120","display_name":"Sibel Koparal","orcid":"https://orcid.org/0000-0001-9574-9652"},"institutions":[{"id":"https://openalex.org/I131835042","display_name":"Bursa Uluda\u011f \u00dcni\u0307versi\u0307tesi\u0307","ror":"https://ror.org/03tg3eb07","country_code":"TR","type":"education","lineage":["https://openalex.org/I131835042"]}],"countries":["TR"],"is_corresponding":true,"raw_author_name":"Sibel Koparal","raw_affiliation_strings":["Department of Mathematics, Bursa Uluda\u011f University, 16059 Bursa, T\u00fcrkiye"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Bursa Uluda\u011f University, 16059 Bursa, T\u00fcrkiye","institution_ids":["https://openalex.org/I131835042"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5019477209","display_name":"Ne\u015fe \u00d6m\u00fcr","orcid":"https://orcid.org/0000-0002-3972-9910"},"institutions":[{"id":"https://openalex.org/I51826884","display_name":"Kocaeli \u00dcniversitesi","ror":"https://ror.org/0411seq30","country_code":"TR","type":"education","lineage":["https://openalex.org/I51826884"]}],"countries":["TR"],"is_corresponding":false,"raw_author_name":"Ne\u015fe \u00d6m\u00fcr","raw_affiliation_strings":["Department of Mathematics, Kocaeli University, 41380 \u0130zmit Kocaeli, T\u00fcrkiye"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Kocaeli University, 41380 \u0130zmit Kocaeli, T\u00fcrkiye","institution_ids":["https://openalex.org/I51826884"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5120292085","display_name":"Sezer Boz","orcid":null},"institutions":[{"id":"https://openalex.org/I131835042","display_name":"Bursa Uluda\u011f \u00dcni\u0307versi\u0307tesi\u0307","ror":"https://ror.org/03tg3eb07","country_code":"TR","type":"education","lineage":["https://openalex.org/I131835042"]}],"countries":["TR"],"is_corresponding":false,"raw_author_name":"Sezer Boz","raw_affiliation_strings":["Department of Mathematics, Bursa Uluda\u011f University, 16059 Bursa, T\u00fcrkiye"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Bursa Uluda\u011f University, 16059 Bursa, T\u00fcrkiye","institution_ids":["https://openalex.org/I131835042"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5075390658","display_name":"Khidir Shaib Mohamed","orcid":"https://orcid.org/0000-0001-8309-6121"},"institutions":[{"id":"https://openalex.org/I156216236","display_name":"Qassim University","ror":"https://ror.org/01wsfe280","country_code":"SA","type":"education","lineage":["https://openalex.org/I156216236"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Khidir Shaib Mohamed","raw_affiliation_strings":["Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia","Department of Mathematics, Qassim University, Buraydah 51452, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0001-8309-6121","affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia","institution_ids":["https://openalex.org/I156216236"]},{"raw_affiliation_string":"Department of Mathematics, Qassim University, Buraydah 51452, Saudi Arabia","institution_ids":["https://openalex.org/I156216236"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5067814181","display_name":"Waseem Ahmad Khan","orcid":"https://orcid.org/0000-0002-4681-9885"},"institutions":[{"id":"https://openalex.org/I138564716","display_name":"Prince Mohammad bin Fahd University","ror":"https://ror.org/03d64na34","country_code":"SA","type":"education","lineage":["https://openalex.org/I138564716"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Waseem Ahmad Khan","raw_affiliation_strings":["Department of Electrical Engineering, Prince Mohammad Bin Fahd University, P.O. Box 1664, Al Khobar 31952, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0002-4681-9885","affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering, Prince Mohammad Bin Fahd University, P.O. Box 1664, Al Khobar 31952, Saudi Arabia","institution_ids":["https://openalex.org/I138564716"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5076494773","display_name":"Alawia Adam","orcid":null},"institutions":[{"id":"https://openalex.org/I156216236","display_name":"Qassim University","ror":"https://ror.org/01wsfe280","country_code":"SA","type":"education","lineage":["https://openalex.org/I156216236"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Alawia Adam","raw_affiliation_strings":["Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia","Department of Mathematics, Qassim University, Buraydah 51452, Saudi Arabia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia","institution_ids":["https://openalex.org/I156216236"]},{"raw_affiliation_string":"Department of Mathematics, Qassim University, Buraydah 51452, Saudi Arabia","institution_ids":["https://openalex.org/I156216236"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5050170120"],"corresponding_institution_ids":["https://openalex.org/I131835042"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":2.7872,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.91217597,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":95,"max":97},"biblio":{"volume":"17","issue":"11","first_page":"1898","last_page":"1898"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12516","display_name":"Advanced Mathematical Theories and Applications","score":0.6256999969482422,"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":0.6256999969482422,"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":0.2736999988555908,"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":0.06260000169277191,"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":0.859000027179718},{"id":"https://openalex.org/keywords/fibonacci-polynomials","display_name":"Fibonacci polynomials","score":0.7601000070571899},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.5665000081062317},{"id":"https://openalex.org/keywords/algebraic-structure","display_name":"Algebraic structure","score":0.5256999731063843},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.5115000009536743},{"id":"https://openalex.org/keywords/distribution","display_name":"Distribution (mathematics)","score":0.44359999895095825},{"id":"https://openalex.org/keywords/orthogonal-polynomials","display_name":"Orthogonal polynomials","score":0.41690000891685486},{"id":"https://openalex.org/keywords/symmetric-polynomial","display_name":"Symmetric polynomial","score":0.3822999894618988},{"id":"https://openalex.org/keywords/algebraic-properties","display_name":"Algebraic properties","score":0.3747999966144562}],"concepts":[{"id":"https://openalex.org/C173734053","wikidata":"https://www.wikidata.org/wiki/Q47577","display_name":"Fibonacci number","level":2,"score":0.859000027179718},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7994999885559082},{"id":"https://openalex.org/C79117305","wikidata":"https://www.wikidata.org/wiki/Q974295","display_name":"Fibonacci polynomials","level":4,"score":0.7601000070571899},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5665000081062317},{"id":"https://openalex.org/C182419690","wikidata":"https://www.wikidata.org/wiki/Q205464","display_name":"Algebraic structure","level":2,"score":0.5256999731063843},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.5115000009536743},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4546000063419342},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.44359999895095825},{"id":"https://openalex.org/C10628310","wikidata":"https://www.wikidata.org/wiki/Q619458","display_name":"Orthogonal polynomials","level":2,"score":0.41690000891685486},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.397599995136261},{"id":"https://openalex.org/C52153879","wikidata":"https://www.wikidata.org/wiki/Q930499","display_name":"Symmetric polynomial","level":4,"score":0.3822999894618988},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3822999894618988},{"id":"https://openalex.org/C2989065042","wikidata":"https://www.wikidata.org/wiki/Q7378971","display_name":"Algebraic properties","level":2,"score":0.3747999966144562},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.36629998683929443},{"id":"https://openalex.org/C52537462","wikidata":"https://www.wikidata.org/wiki/Q5282055","display_name":"Discrete orthogonal polynomials","level":3,"score":0.3555000126361847},{"id":"https://openalex.org/C21736991","wikidata":"https://www.wikidata.org/wiki/Q17006917","display_name":"Classical orthogonal polynomials","level":3,"score":0.3537999987602234},{"id":"https://openalex.org/C201292218","wikidata":"https://www.wikidata.org/wiki/Q740970","display_name":"Recurrence relation","level":2,"score":0.3465000092983246},{"id":"https://openalex.org/C86607863","wikidata":"https://www.wikidata.org/wiki/Q5275279","display_name":"Difference polynomials","level":3,"score":0.3312999904155731},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.2985000014305115},{"id":"https://openalex.org/C18912844","wikidata":"https://www.wikidata.org/wiki/Q4724001","display_name":"Algebraic expression","level":3,"score":0.2858999967575073},{"id":"https://openalex.org/C171078966","wikidata":"https://www.wikidata.org/wiki/Q111029","display_name":"Root (linguistics)","level":2,"score":0.28189998865127563},{"id":"https://openalex.org/C126385604","wikidata":"https://www.wikidata.org/wiki/Q1087609","display_name":"Chromatic polynomial","level":3,"score":0.27000001072883606},{"id":"https://openalex.org/C3634175","wikidata":"https://www.wikidata.org/wiki/Q2503280","display_name":"Lucas number","level":3,"score":0.265500009059906},{"id":"https://openalex.org/C174757984","wikidata":"https://www.wikidata.org/wiki/Q4351325","display_name":"Pisano period","level":5,"score":0.25760000944137573},{"id":"https://openalex.org/C171686973","wikidata":"https://www.wikidata.org/wiki/Q17098676","display_name":"Macdonald polynomials","level":4,"score":0.2547000050544739},{"id":"https://openalex.org/C42321380","wikidata":"https://www.wikidata.org/wiki/Q849705","display_name":"Characteristic polynomial","level":3,"score":0.2531999945640564}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/sym17111898","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17111898","pdf_url":"https://www.mdpi.com/2073-8994/17/11/1898/pdf","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:0deeeebb-c64a-459e-a1f1-677e3a60f2ba","is_oa":false,"landing_page_url":"https://avesis.uludag.edu.tr/publication/details/0deeeebb-c64a-459e-a1f1-677e3a60f2ba/oai","pdf_url":null,"source":{"id":"https://openalex.org/S7407055210","display_name":"Bursa Uludag University - AVESIS","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":null,"raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"doi:10.3390/sym17111898","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17111898","pdf_url":"https://www.mdpi.com/2073-8994/17/11/1898/pdf","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/G2186466424","display_name":null,"funder_award_id":"QU-APC-2025","funder_id":"https://openalex.org/F4320324403","funder_display_name":"Qassim University"}],"funders":[{"id":"https://openalex.org/F4320324403","display_name":"Qassim University","ror":"https://ror.org/01wsfe280"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4415987086.pdf","grobid_xml":"https://content.openalex.org/works/W4415987086.grobid-xml"},"referenced_works_count":12,"referenced_works":["https://openalex.org/W68100126","https://openalex.org/W1895808807","https://openalex.org/W1980773437","https://openalex.org/W2030221505","https://openalex.org/W2039482973","https://openalex.org/W2074239094","https://openalex.org/W2113956435","https://openalex.org/W4212799367","https://openalex.org/W4327598396","https://openalex.org/W4402832191","https://openalex.org/W4403205658","https://openalex.org/W4403680689"],"related_works":[],"abstract_inverted_index":{"This":[0],"study":[1],"presents":[2],"a":[3,23,107,131],"unified":[4],"framework":[5],"for":[6],"the":[7,51,65,74,103,115,118,122,125,135,139,141,146],"simultaneous":[8],"analysis":[9,28],"of":[10,20,43,50,80,106,109,117,134,138,149],"generalized":[11,33,52],"Fibonacci":[12,34,53],"numbers":[13],"and":[14,29,56,70,77,120,145],"their":[15,91,112],"associated":[16],"polynomial":[17,84],"extensions,":[18],"both":[19],"which":[21],"play":[22],"significant":[24],"role":[25],"in":[26],"combinatorial":[27],"discrete":[30],"mathematics.":[31],"The":[32],"polynomials":[35,54,68],"have":[36,95],"been":[37,96],"extended":[38,66],"to":[39,101,129],"four":[40],"new":[41],"families":[42],"polynomials,":[44,110,140],"each":[45],"defined":[46,83],"through":[47],"systematic":[48],"extensions":[49],"Ukl(\u03c2)":[55],"Vkl(\u03c2).":[57],"In":[58],"addition,":[59],"we":[60,99],"explore":[61],"further":[62],"generalizations":[63],"involving":[64],"Humbert-type":[67],"Ukl,m(r)(\u03c2)":[69],"Vkl,m(r)(\u03c2).":[71],"Based":[72],"on":[73],"algebraic":[75,87],"structure":[76],"generating":[78],"functions":[79],"these":[81],"newly":[82],"families,":[85],"several":[86],"identities":[88],"that":[89],"reveal":[90],"rich":[92],"mathematical":[93],"properties":[94,137],"derived.":[97],"Additionally,":[98],"aim":[100],"present":[102],"graphical":[104],"representations":[105],"family":[108],"analyze":[111],"roots,":[113,119],"examine":[114],"distribution":[116,144,148],"investigate":[121],"correlations":[123],"among":[124],"largest":[126],"roots.":[127],"Finally,":[128],"gain":[130],"deeper":[132],"understanding":[133],"structural":[136],"root":[142,150],"magnitude":[143],"density":[147],"values":[151],"are":[152],"also":[153],"examined.":[154]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2}],"updated_date":"2026-06-24T13:16:06.693445","created_date":"2025-11-07T00:00:00"}