Difference between revisions of "Fibonacci polynomial"

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=Properties=
 
=Properties=
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<strong>Theorem:</strong> The following formula holds:
 
$$\displaystyle\sum_{k=0}^{\infty} F_k(x)t^n = \dfrac{t}{1-xt-t^2},$$
 
where $F_k$ denotes a [[Fibonacci polynomial]].
 
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<strong>Proof:</strong> █
 
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=References=
<strong>Theorem:</strong> The following formula holds:
 
$$F_{-n}(x)=(-1)^{n-1}F_n(x).$$
 
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<strong>Proof:</strong> █
 
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[[Category:SpecialFunction]]
<strong>Theorem:</strong> The following formula holds:
 
$$F_{n+1}(x)F_{n-1}(x)-F_n(x)^2=(-1)^n.$$
 
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<strong>Proof:</strong> █
 
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<strong>Theorem:</strong> The following formula holds:
 
$$F_{2n}(x)=F_n(x)L_n(x),$$
 
where $F_n$ denotes a [[Fibonacci polynomial]] and $L_n$ denotes a [[Lucas polynomial]].
 
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<strong>Proof:</strong> █
 
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Latest revision as of 23:23, 27 June 2016

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.

Properties

References