Generalized Fibonacci Polynomials and Their Properties

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(ς) and Vkl(ς). In addition, we explore further generalizations involving the extended Humbert-type polynomials Ukl,m(r)(ς) and Vkl,m(r)(ς). 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.