{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,23]],"date-time":"2026-06-23T17:00:12Z","timestamp":1782234012449,"version":"3.54.5"},"reference-count":18,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T00:00:00Z","timestamp":1762473600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Graduate Studies and Scientific Research at Qassim University","award":["QU-APC-2025"],"award-info":[{"award-number":["QU-APC-2025"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This study presents a unified framework for the simultaneous analysis of generalized Fibonacci numbers and their associated polynomial extensions, both of which play a significant role in combinatorial analysis and discrete mathematics. The generalized Fibonacci polynomials have been extended to four new families of polynomials, each defined through systematic extensions of the generalized Fibonacci polynomials Ukl(\u03c2) and Vkl(\u03c2). In addition, we explore further generalizations involving the extended Humbert-type polynomials Ukl,m(r)(\u03c2) and Vkl,m(r)(\u03c2). Based on the algebraic structure and generating functions of these newly defined polynomial families, several algebraic identities that reveal their rich mathematical properties have been derived. Additionally, we aim to present the graphical representations of a family of polynomials, analyze their roots, examine the distribution of the roots, and investigate the correlations among the largest roots. Finally, to gain a deeper understanding of the structural properties of the polynomials, the root magnitude distribution and the density distribution of root values are also examined.<\/jats:p>","DOI":"10.3390\/sym17111898","type":"journal-article","created":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T09:14:45Z","timestamp":1762506885000},"page":"1898","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Generalized Fibonacci Polynomials and Their Properties"],"prefix":"10.3390","volume":"17","author":[{"given":"Sibel","family":"Koparal","sequence":"first","affiliation":[{"name":"Department of Mathematics, Bursa Uluda\u011f University, 16059 Bursa, T\u00fcrkiye"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ne\u015fe","family":"\u00d6m\u00fcr","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Kocaeli University, 41380 \u0130zmit Kocaeli, T\u00fcrkiye"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sezer","family":"Boz","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Bursa Uluda\u011f University, 16059 Bursa, T\u00fcrkiye"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8309-6121","authenticated-orcid":false,"given":"Khidir Shaib","family":"Mohamed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4681-9885","authenticated-orcid":false,"given":"Waseem Ahmad","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, Prince Mohammad Bin Fahd University, P.O. Box 1664, Al Khobar 31952, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Alawia","family":"Adam","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1080\/00150517.1996.12429098","article-title":"Extension of a synthesis for a class of polynomial sequences","volume":"34","author":"Horadam","year":"1996","journal-title":"Fibonacci Quart."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3179","DOI":"10.1016\/j.chaos.2009.04.048","article-title":"On generalized Fibonacci and Lucas polynomials","volume":"42","author":"Haukkanen","year":"2009","journal-title":"Chaos Solitons Fractals"},{"key":"ref_3","first-page":"204","article-title":"Generalized Humbert polynomials via generalized Fibonacci polynomials","volume":"307","author":"Wang","year":"2017","journal-title":"Appl. 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