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

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.