Fibonacci polynomials

Fibonacci polynomials are defined recursively as follow

F_{n}(x):=\\left\\{\\begin{array}[]{ll}1&{\\rm if~{}}n=1;\\\\x&{\\rm if~{}}n=2;\\\\xF_{n-1}(x)+F_{n-2}(x),&{\\rm if~{}}n>2.\\end{array}\\right.

The main (and obvious) property of this polynomials is that $F_{n}(1)$ is $n^{\\rm th}$ number in Fibonacci sequence.

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