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Fibonacci Polynomial


The $W$ Polynomials obtained by setting $p(x)=x$ and $q(x)=1$ in the Lucas Polynomial Sequence. (The corresponding $w$ Polynomials are called Lucas Polynomials.) The Fibonacci polynomials are defined by the Recurrence Relation

\end{displaymath} (1)

with $F_1(x)=1$ and $F_2(x)=x$. They are also given by the explicit sum formula
F_n(x)=\sum_{j=0}^{\left\lfloor{(n-1)/2}\right\rfloor }{n-j-1\choose j}x^{n-2j-1},
\end{displaymath} (2)

where $\left\lfloor{x}\right\rfloor $ is the Floor Function and ${n\choose m}$ is a Binomial Coefficient. The first few Fibonacci polynomials are
$\displaystyle F_1(x)$ $\textstyle =$ $\displaystyle 1$  
$\displaystyle F_2(x)$ $\textstyle =$ $\displaystyle x$  
$\displaystyle F_3(x)$ $\textstyle =$ $\displaystyle x^2+1$  
$\displaystyle F_4(x)$ $\textstyle =$ $\displaystyle x^3+2x$  
$\displaystyle F_5(x)$ $\textstyle =$ $\displaystyle x^4+3x^2+1.$  

The Fibonacci polynomials are normalized so that
\end{displaymath} (3)

where the $F_n$s are Fibonacci Numbers.

The Fibonacci polynomials are related to the Morgan-Voyce Polynomials by

$\displaystyle F_{2n+1}(x)$ $\textstyle =$ $\displaystyle b_n(x^2)$ (4)
$\displaystyle F_{2n+n2}(x)$ $\textstyle =$ $\displaystyle x B_n(x^2)$ (5)

(Swamy 1968).

See also Brahmagupta Polynomial, Fibonacci Number, Morgan-Voyce Polynomial


Swamy, M. N. S. ``Further Properties of Morgan-Voyce Polynomials.'' Fib. Quart. 6, 167-175, 1968.

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