*first type* or *Lucas type* and it is of *Second type* or *Fibonacci type* if it satisfies the Binet formula (\[bineformulauno\]). Horadam [@horadam-synthesis] and André-Jeannin [@Richard] have studied these polynomials in detail.

The sequence of polynomials that have Binet representations $R_n(x)$ or $L_n(x)$ depend only on $d(x)$ and $g(x)$ defined on the generalized Fibonacci polynomials. We say that a generalized Fibonacci sequence of Lucas (Fibonacci) type is *equivalent* to a sequence of the Fibonacci (Lucas) type, if their recursive sequences are determined by the same polynomials $d(x)$ and $g(x)$. Notice that two equivalent polynomials have the same $a(x)$ and $b(x)$ in their Binet representations.

For example, the Lucas polynomial is a GFP of Lucas type, whereas the Fibonacci polynomial is a GFP of Fibonacci type. Lucas and Fibonacci polynomials are equivalent because $d(x)=x$ and $g(x)=1$ (see Table \[familiarfibonacci\]). Note that in their Binet representations they both have $a(x)= (x+\sqrt{x^2+4})/2$ and $b(x)=(x-\sqrt{x^2+4})/2$. The Table \[equivalent\] is based on information from the following papers [@Richard; @florezHiguitaMuk; @horadam-synthesis]. The leftmost polynomials in Table \[equivalent\] are of the Lucas type and their equivalent polynomials are in the third column on the same line. In the last two columns of Table \[equivalent\] we can see 