(1). Since $ G_{n}^{*}(x)$ is a Fibonacci polynomial of the Lucas type, we have that $G_{n}^{*}(x)$ satisfies the Binet formula $L_n(x)= (a^{n}+b^{n})/\alpha$ given in (\[bineformulados\]). Therefore, $$\label{divisity:Hogat:property2:formula1}
G_{m+1}^{*}(x)G_{n+1}^{*}(x)+g(x)G_{m}^{*}(x)G_{n}^{*}(x)$$ is equal to, $$\left[a^{n+1}+b^{n+1}\right]\left[a^{m+1}+b^{m+1}\right]+g(x)\left[a^{n}+b^{n}\right]\left[a^{m}+b^{m}\right].$$ Simplifying and factoring we see that this expression is equal to $$a^{m+n}\left[a^2+g(x)\right]+b^{m+n}\left[b^2+g(x)\right]+(ab+g(x))\left[a^mb^n+a^nb^m\right].$$ Since $$ab=-g(x),\; a^2+g(x)=a(a-b),\text{ and } b^2+g(x)=-b(a-b),$$ we have that the expression in (\[divisity:Hogat:property2:formula1\]) is equal to $(a-b)(a^{m+n+1}-b^{m+n+1})$. We recall that $G_{m+n+1}^{'}(x)$ is equivalent to $G_{m+n+1}^{*}(x)$. Thus, $G^{'}_{m+n+1}(x) =(a^{m+n+1}-b^{m+n+1})/(a-b)$. Therefore, $(a-b)^2G^{'}_{m+n+1}(x) =(a-b)\left[a^{m+n+1}-b^{m+n+1}\right]$. This completes the proof of part (1).

Proof of part (2). From the proof of part (1) we know that $$(a-b)^2G^{'}_{m+n+1}(x)=(a-b)[a^{m+n+1}-b^{m+n+1}].$$ Simplifying the right side of the previous equality we have that $$(a-b)^2G^{'}_{m+n+1}(x)=a^{m+n+2}-ba^{m+n+1}- ab^{m+n+1} + b^{m+n+2}.$$ So, $(a-b)^2G^{'}_{m+n+1}(x)=a^{m+n+2}+ b^{m+n+2} -ab[a^{m+n}+ b^{m+n}]$. We recall that $ab=-g(x)$. Thus, $$(a-b)^2G^{'}_{m+n+1}(x)=a^{m+n+2}+ b^{m+n+2} + g(x)[a^{m+n}+ b^{m+n}].$$ This and the Binet formula (\[bineformulados\]), imply that $$(a-b)^2G^{'}_{m+n+1}(x)=G_{m+n+2}^{*}(x) + g(x)G_{m+n}^{*}(x).$$ So, the proof follows from part (1) of this Proposition.

\[Dic2\] Let $\{ G_{n}^{*}(x)\}$ be a GFP sequence of the Lucas type. If $m$, $n$, $r$, and $q$ are positive integers, then

(1) if $m\leq n$, then $G_{m+n}^{*}(x)=\alpha G_{m}^{*}(x)G_{n}^{*}(x)+(-1)^{m+1}(g(x))^mG_{n-m}^{*}(x)$,

(2) if $r<m$, then there is a polynomial $T(x)$ such that $$G_{mq+r}^{*}(x)=
        \left\{
             \begin{array}{ll}
                G_{m}^{*}(x)T(x)+(-1)^{m(t-1)+t+r}(g(x))^{(t-1)m+r} G_{m-r}^{*}(x), & \hbox{ if } $q$ \hbox{ is odd;} 