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| | =Properties= | | =Properties= |
| − | <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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| − | $$\displaystyle\sum_{k=0}^{\infty} F_k(x)t^n = \dfrac{t}{1-xt-t^2},$$
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| − | 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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| − |
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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">
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| − | <strong>Proof:</strong> █
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| − | </div>
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| − | </div>
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| − |
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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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| − | $$F_{n+1}(x)F_{n-1}(x)-F_n(x)^2=(-1)^n.$$
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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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| − |
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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]].
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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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| | =References= | | =References= |
| − | * {{PaperReference|Sur la série des inverse de nombres de Fibonacci|1899|Edmund Landau|next=findme}}
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| | [[Category:SpecialFunction]] | | [[Category:SpecialFunction]] |
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
