two types, the Lucas type and the Fibonacci type, depending on their closed formulas or their Binet formulas (see for example, $L_n(x)$ (\[bineformulados\]) and $R_n(x)$ (\[bineformulauno\]), and the Table \[equivalent\]). That is, if after solving the characteristic polynomial of the GFP we obtain a closed formula that look like the Binet formula for Fibonacci (Lucas) numbers, it is called Fibonacci (Lucas) type polynomials. Familiar examples of Fibonacci type polynomials are: Fibonacci polynomials, Pell polynomials, Fermat polynomials, Chebyshev polynomials of second kind, Jacobsthal polynomials and one type of Morgan-Voyce polynomials. Examples of Lucas type polynomials are: Lucas polynomials, Pell-Lucas polynomials, Fermat-Lucas polynomials, Chebyshev polynomials of first kind, Jacobsthal-Lucas polynomials and second type of Morgan-Voyce polynomials.

In Theorem \[gcd:property:fibonacci\] we prove that a GFP satisfies the strong divisibility property if and only if it is of Fibonacci type. The Theorem \[second:main:thm\] shows that the Lucas type polynomials satisfy the strong divisibility property partially and also gives the $gcd$ for those cases in which the property is not satisfied.

A Lucas type polynomial is equivalent to a Fibonacci type polynomial if they both have the same recurrence relation but different initial conditions (see also Flórez et al. [@florezHiguitaMuk]). The Theorem \[combine:gcd:Lucas:Fibobacci\] proves that 