we present some identities that the GFP satisfy. These identities are required for the proofs of certain divisibility properties of the GFP. The results in this section are proved using the Binet formulas (see Section \[General:Fibonacci:Polynomial\]). Proposition \[divisity:Hogat:property1\] part (1) is a variation of a result proved by Hoggatt and Long [@HoggattLong], similarly Proposition \[divisity:property:fibonacci\] is a variation of a divisibility property proved by them in the same paper.

In 1963 Ruggles [@koshy] proved that $F_{n+m}= F_{n} L_{m}-(-1)^m F_{n-m}$. Proposition \[divisity:Hogat:property1\] parts (2) and (3) are a generalization of this numerical identity. In 1972 Hansen [@hansen] proved that $ 5 F_{m +n - 1}=L_{m} L_{n} + L_{m - 1} L_{n -1} $. Proposition \[divisity:Hogat:property2\] part (1) is a generalization of this numerical identity.

\[divisity:Hogat:property1\] If $\{ G_{n}^{*}(x)\}$ and $\{ G_{n}^{'}(x)\}$ are equivalent GFP sequences, then

(1) $G_{m+n+1}^{'}(x)= G_{m+1}^{'}(x)G_{n+1}^{'}(x)+g(x)G_{m}^{'}(x)G_{n}^{'}(x)$,

(2) if $n\ge m$, then $G_{n+m}^{'}(x)= \alpha G_{n}^{'}(x) G_{m}^{*}(x)-(-g(x))^{m} G_{n-m}^{'}(x)$,

(3) if $n\ge m$, then $G_{n+m}^{'}(x)= \alpha G_{m}^{'}(x)G_{n}^{*}(x) +(-g(x))^{m} G_{n-m}^{'}(x)$.

We prove part (1). We know that $\{ G_{n}^{'}(x)\}$ satisfies the Binet formula (\[bineformulauno\]). That is, $R_n(x)= (a^{n}-b^{n})/(a-b)$ . (Recall that we use $a:=a(x)$ and $b:=b(x)$.)

Therefore, $G_{m+1}^{'}(x)G_{n+1}^{'}(x)+g(x)G_{m}^{'}(x)G_{n}^{'}(x)$ is equal to, $$\left[(a^{m+1}-b^{m+1})(a^{n+1}-b^{n+1})+g(x)(a^{m}-b^{m})(a^{n}-b^{n})\right]/(a-b)^{2}.$$ Simplifying and factoring terms we obtain, $$\left[\left(a^{n+m}(a^2+g(x))+b^{n+m}(b^2+g(x))\right)-(a^nb^m+b^na^m)(ab+g(x))\right]/(a-b)^{2}.$$ Next, since $ab=-g(x)$, 