Revision as of 17:51, 21 January 2008 by imported>David E. Volk
In mathematics, Fibonacci polynomials are a generalization of Fibonacci numbers. These polynomials are defined by:
![{\displaystyle F_{n}(x)=\left\{{\begin{matrix}1,\qquad \qquad \qquad \qquad &{\mbox{if }}n=1\\x,\qquad \qquad \qquad \qquad &{\mbox{if }}n=2\\xF_{n-1}(x)+F_{n-2}(x),&{\mbox{if }}n\geq 3\end{matrix}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35cbcac6f2b787b374ddcab3d597d2ec8132a5dd)
The first few Fibonacci polynomials are:
![{\displaystyle F_{1}(x)=1\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/658facd8ef73879e20b3143b4615c4b9f749b26d)
![{\displaystyle F_{2}(x)=x\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a29f80479602de8440e796c9d6224d1c7cde2e09)
![{\displaystyle F_{3}(x)=x^{2}+1\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ef6aa18ca8d576b4329459ea51eb3dd67bca70d)
![{\displaystyle F_{4}(x)=x^{3}+2x\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/764aadf16474f43df83cc768073d216036d545d1)
![{\displaystyle F_{5}(x)=x^{4}+3x^{2}+1\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93f6dcfb45e870d88d29ba6ef9fb7c28106438bd)
![{\displaystyle F_{6}(x)=x^{5}+4x^{3}+3x\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31b06bda879fee7755b9a905a1aa27b2e7131744)
The Fibonacci numbers are recovered by evaluating the polynomials at x = 1.
See also