Difference between revisions of "Fibonacci polynomial"

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m (Tom moved page Fibonacci to Fibonacci polynomial over redirect)
(Properties)
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$$\displaystyle\sum_{k=0}^{\infty} F_k(x)t^n = \dfrac{t}{1-xt-t^2},$$
 
$$\displaystyle\sum_{k=0}^{\infty} F_k(x)t^n = \dfrac{t}{1-xt-t^2},$$
 
where $F_k$ denotes a [[Fibonacci polynomial]].
 
where $F_k$ denotes a [[Fibonacci polynomial]].
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<div class="mw-collapsible-content">
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<strong>Proof:</strong> █
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</div>
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</div>
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 +
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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<strong>Theorem:</strong> The following formula holds:
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$$F_{-n}(x)=(-1)^{n-1}F_n(x).$$
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<div class="mw-collapsible-content">
 +
<strong>Proof:</strong> █
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</div>
 +
</div>
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 +
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 +
<strong>Theorem:</strong> The following formula holds:
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$$F_{n+1}(x)F_{n-1}(x)-F_n(x)^2=(-1)^n.$$
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<div class="mw-collapsible-content">
 +
<strong>Proof:</strong> █
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</div>
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</div>
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 +
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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<strong>Theorem:</strong> The following formula holds:
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$$F_{2n}(x)=F_n(x)L_n(x),$$
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where $F_n$ denotes a [[Fibonacci polynomial]] and $L_n$ denotes a [[Lucas polynomial]].
 
<div class="mw-collapsible-content">
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █  
 
<strong>Proof:</strong> █  
 
</div>
 
</div>
 
</div>
 
</div>

Revision as of 22:59, 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.

Properties

Theorem: 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.

Proof:

Theorem: The following formula holds: $$F_{-n}(x)=(-1)^{n-1}F_n(x).$$

Proof:

Theorem: The following formula holds: $$F_{n+1}(x)F_{n-1}(x)-F_n(x)^2=(-1)^n.$$

Proof:

Theorem: 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.

Proof: