Editing Fibonacci polynomials (section)

== Properties ==
* The degree of ''F''<sub>''n''</sub> is ''n''&nbsp;&minus;&nbsp;1 and the degree of ''L''<sub>''n''</sub> is ''n''.  

* The Fibonacci and Lucas numbers are recovered by evaluating the polynomials at ''x''&nbsp;=&nbsp;1;  [[Pell numbers]] are recovered by evaluating ''F''<sub>''n''</sub> at ''x''&nbsp;=&nbsp;2.  

* The [[Generating function#Ordinary generating function|ordinary generating functions]] for the sequences are:<ref>{{MathWorld | urlname=FibonacciPolynomial | title=Fibonacci Polynomial}}</ref>
*:<math> \sum_{n=0}^\infty F_n(x) t^n = \frac{t}{1-xt-t^2}</math>
*:<math> \sum_{n=0}^\infty L_n(x) t^n = \frac{2-xt}{1-xt-t^2}.</math>

*The polynomials can be expressed in terms of [[Lucas sequence]]s as 
*:<math>F_n(x) = U_n(x,-1),\,</math>
*:<math>L_n(x) = V_n(x,-1).\,</math>

*They can also be expressed in terms of [[Chebyshev polynomials]] <math>\mathcal{T}_n(x)</math> and <math>\mathcal{U}_n(x)</math> as
*:<math>F_n(x) = i^{n-1}\cdot\mathcal{U}_{n-1}(\tfrac{-ix}2),\,</math>
*:<math>L_n(x) = 2\cdot i^n\cdot\mathcal{T}_n(\tfrac{-ix}2),\,</math>
:where <math>i</math> is the [[imaginary unit]].