the $a(x)$ and $b(x)$ that the pairs of equivalent polynomials share. It is easy to obtain the characteristic equations of the sequences given in Table \[familiarfibonacci\], and the roots of the equations are $a(x)$ and $b(x)$.

For the sake of simplicity throughout this paper we use $a$ in place of $a(x)$ and $b$ in place of $b(x)$ when they appear in the Binet formulas. We use the notation $G_n^{*}$ or $G_n^{'}$ for $G_n$ depending on when it satisfies the Binet formulas (\[bineformulados\]) or (\[bineformulauno\]), respectively, (see Section \[mainresults:section\]).

For most of the proofs of GFP of Lucas type it is required that $\gcd(p_0(x), p_1(x))=1$, $\gcd(p_0(x), d(x))=1$ and $\gcd(d(x), g(x))=1$. It is easy to see that $\gcd(\alpha, G_{n}^{*}(x))=1$. Therefore, for the rest the paper we suppose that these four mentioned conditions hold for all generalized Fibonacci polynomial sequences of Lucas type treated here. We use $\rho$ to denote $\gcd(d(x),G_1(x))$. Notice that in the definition Pell-Lucas we have that $p_0(x)=2$ and $p_1(x)=2x$. Thus, the $\gcd(p_0(x),p_1(x))\ne 1$. Therefore, Pell-Lucas does not satisfy the extra conditions that were just imposed for Generalized Fibonacci polynomial. To solve this problem we define *Pell-Lucas-prime* as follows: $$Q_0^{\prime}(x)= 1, \; Q_1^{\prime}(x)= x, \; \text{and} \;  Q_{n}^{\prime}(x)= 2x Q_{n 