@article{Du2024, 
author = {Tingting Du and Zhengang Wu},
title = {Some identities of the generalized bi-periodic Fibonacci and Lucas polynomials},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {3},
pages = {7492-7510},
keywords = {generalized bi-periodic Fibonacci polynomial, generalized bi-periodic Lucas polynomial, Binet formula, matrix theory},
url = {https://www.sciopen.com/article/10.3934/math.2024363},
doi = {10.3934/math.2024363},
abstract = {In this paper, we considered the generalized bi-periodic Fibonacci polynomials, and obtained some identities related to generalized bi-periodic Fibonacci polynomials using the matrix theory. In addition, the generalized bi-periodic Lucas polynomial was defined by        L          n            (    x    )    =  b  p      (    x    )        L          n      −      1            (    x    )    +  q      (    x    )        L          n      −      2            (    x    )    (if    n is even) or        L          n            (    x    )    =  a  p      (    x    )        L          n      −      1            (    x    )    +  q      (    x    )        L          n      −      2            (    x    )    (if    n is odd), with initial conditions        L          0            (    x    )    =  2,        L          1            (    x    )    =  a  p      (    x    )  , where    p      (    x    )   and    q      (    x    )   were nonzero polynomials in    Q      [    x    ]  . We obtained a series of identities related to the generalized bi-periodic Fibonacci and Lucas polynomials.}
}