40-00-FibonacciPolynomials.tex

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\pmtitle{Fibonacci polynomials}
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\pmtype{Definition}
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%\pmkeywords{Fibonacci sequence}

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\begin{document}
Fibonacci polynomials are defined recursively as follow
$$
F_n(x):=
\left\{
 \begin{array}{ll}
  1 & {\rm if~} n=1;\\
  x & {\rm if~} n=2;\\
  x F_{n-1}(x)+F_{n-2}(x), & {\rm if~} n>2.
 \end{array}
\right.
$$
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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