Difference between revisions of "Fibonacci polynomial"
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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]]. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$F_{-n}(x)=(-1)^{n-1}F_n(x).$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$F_{n+1}(x)F_{n-1}(x)-F_n(x)^2=(-1)^n.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <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]]. | ||
<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: █
