we let $p_0(x)=0$, $p_1(x)=1$, $d(x)=x$, and $g(x)=1$ we obtain the regular Fibonacci polynomial sequence. Thus, $$F_0(x)= 0, \; F_1(x)= 1, \; \text{and} \;  F_{n}(x)= x F_{n - 1}(x) + F_{n - 2}(x) \text{ for } n\ge 2.$$

Letting $x=1$ and choosing the correct values for $p_0(x)$, $p_1(x)$, $d(x)$ and $g(x)$, the generalized Fibonacci polynomial sequence gives rise to three classical numerical sequences, the Fibonacci sequence, the Lucas sequence and the generalized Fibonacci sequence.

In Table \[familiarfibonacci\] there are more familiar examples of GFP (see [@florezHiguitaMuk; @Pell; @Fermat; @koshy]). Hoggatt and Bicknell-Johnson [@hoggatt] show that Schechter polynomials are another example of generalized Fibonacci polynomials.

\[!ht\]

Fibonacci type and Lucas type polynomials
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If we impose some conditions on the Definition (\[Fibonacci;general\]) we obtain two type of distinguishable polynomials. We say that a sequences as in (\[Fibonacci;general\]) is *Lucas type* or *first type* if $2p_{1}(x)=p_{0}(x)d(x)$ with $p_{0}\ne 0$. We say that a sequences as in (\[Fibonacci;general\]) is *Fibonacci type* or *second type* if $p_{0}(x)=0$ with $p_{1}(x)$ a constant.

If $d^2(x)+4g(x)> 0$, then the explicit formula for the recurrence relation (\[Fibonacci;general\]) is given by

$$\label{solutionrecurrencerelationuno}
 G_{n}(x) = t_1 a^{n} + t_2 b^{n}$$

where $a(x)$ and $b(x)$ are the solutions of the quadratic equation 