Some identities of the generalized bi-periodic Fibonacci and Lucas polynomials

Abstract

<abstract><p>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}\left (x \right) = bp\left (x \right) L_{n-1}\left (x \right)+q\left (x \right)L_{n-2}\left (x \right) $ (if $ n $ is even) or $ L_{n}\left (x \right) = ap\left (x \right) L_{n-1}\left (x \right)+q\left (x \right)L_{n-2}\left (x \right) $ (if $ n $ is odd), with initial conditions $ L_{0}\left (x \right) = 2 $, $ L_{1}\left (x \right) = ap\left (x \right) $, where $ p\left (x \right) $ and $ q\left (x \right) $ were nonzero polynomials in $ Q \left [ x \right ] $. We obtained a series of identities related to the generalized bi-periodic Fibonacci and Lucas polynomials.</p></abstract>