Difference between revisions of "Fibonacci polynomial"

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m (Tom moved page Fibonacci polynomial to Fibonacci)
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xF_{n-1}(x)+F_{n-2}(x)&; n\geq 2.
 
xF_{n-1}(x)+F_{n-2}(x)&; n\geq 2.
 
\end{array} \right.$$
 
\end{array} \right.$$
 +
 +
The first few Fibonacci polynomials are
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$$F_0(x)=1,$$
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$$F_1(x)=1,$$
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$$F_2(x)=x,$$
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$$F_3(x)=x^2+1,$$
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$$F_4(x)=x^3+2x,$$
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$$F_5(x)=x^4+3x^2+1.$$
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Note the similarity with the [[Lucas polynomial|Lucas polynomials]].
 
Note the similarity with the [[Lucas polynomial|Lucas polynomials]].

Revision as of 22:55, 11 April 2015

Fibonacci polynomials are defined by $$F_n(x)=\left\{ \begin{array}{ll} 0&; n=0 \\ 1&; n=1 \\ xF_{n-1}(x)+F_{n-2}(x)&; n\geq 2. \end{array} \right.$$

The first few Fibonacci polynomials are $$F_0(x)=1,$$ $$F_1(x)=1,$$ $$F_2(x)=x,$$ $$F_3(x)=x^2+1,$$ $$F_4(x)=x^3+2x,$$ $$F_5(x)=x^4+3x^2+1.$$

Note the similarity with the Lucas polynomials.