two equivalent GFP satisfies the strong divisibility property partially and gives the $gcd$ for those cases in which the property is not satisfied.

In 1969 Webb and Parberry [@Webb] extended the strong divisibility property to the Fibonacci Polynomials. In 1974 Hoggatt and Long [@HoggattLong] proved the strong divisibility property for one type of generalized Fibonacci polynomial. In 1978 Hoggatt and Bicknell-Johnson [@hoggatt] extended the result mentioned in [@HoggattLong] to some cases of Fibonacci type polynomials. However, they did not prove the necessary and sufficient condition and their paper does not cover the Lucas type case. In 2005 Rayes, Trevisan, and Wang [@Rayes] proved that the strong divisibility property holds partially for the Chebyshev polynomials (we prove the general result in Theorem \[second:main:thm\]). Over the years several other authors have been also interested in the divisibility properties of sequences, some papers are [@hall; @kimberling; @kimberlingStrong1; @kimberlingStrong2; @lucas; @mcdaniel; @norfleet; @ward].

Lucas [@lucas] proved the strong divisibility property (SDP) for Fibonacci numbers. However, the study of SDP for Lucas numbers took until 1991, when McDaniel [@mcdaniel] provided proofs that the Lucas numbers satisfy the SDP partially. In 1995 Hilton et al. [@hilton] gave some more precise results about this property. As mentioned 