Fibonacci polynomials

In mathematics, the Fibonacci polynomials are a polynomial sequence which can be considered as a generalization of the Fibonacci numbers. The polynomials generated in a similar way from the Lucas numbers are called Lucas polynomials.
Definition.
These Fibonacci polynomials are defined by a recurrence relation:
The first few Fibonacci polynomials are:
The Lucas polynomials use the same recurrence with different starting values:
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The first few Lucas polynomials are:
The Fibonacci and Lucas numbers are recovered by evaluating the polynomials at "x" = 1; Pell numbers are recovered by evaluating "F""n" at "x" = 2. The degrees of "F""n" is "n" − 1 and the degree of "L""n" is "n". The ordinary generating function for the sequences are:
The polynomials can be expressed in terms of Lucas sequences as 
Identities.
As particular cases of Lucas sequences, Fibonacci polynomials satisfy a number of identities. 
First, they can be defined for negative indices by
Other identities include:
Closed form expressions, similar to Binet's formula are:
where 
are the solutions (in "t") of
A relationship between the Fibonacci polynomials and the standard basis polynomials is given by 
For example,
A proof of this fact is given starting from page 5 here.
For Lucas Polynomials "n" > 0, we have
Combinatorial interpretation.
If "F"("n","k") is the coefficient of "xk" in "Fn"("x"), so 
then "F"("n","k") is the number of ways an "n"−1 by 1 rectangle can be tiled with 2 by 1 dominoes and 1 by 1 squares so that exactly "k" squares are used. Equivalently, "F"("n","k") is the number of ways of writing "n"−1 as an ordered sum involving only 1 and 2, so that 1 is used exactly "k" times. For example F(6,3)=4 and 5 can be written in 4 ways, 1+1+1+2, 1+1+2+1, 1+2+1+1, 2+1+1+1, as a sum involving only 1 and 2 with 1 used 3 times. By counting the number of times 1 and 2 are both used in such a sum, it is evident that "F"("n","k") is equal to the binomial coefficient
when "n" and "k" have opposite parity. This gives a way of reading the coefficients from Pascal's triangle as shown on the right.
References.
 