Editing Fibonacci polynomials (section)

==Identities==
{{main|Lucas sequence}}
As particular cases of Lucas sequences, Fibonacci polynomials satisfy a number of identities, such as<ref name=Springer/>
:<math>F_{m+n}(x)=F_{m+1}(x)F_n(x)+F_m(x)F_{n-1}(x)\,</math>
:<math>L_{m+n}(x)=L_m(x)L_n(x)-(-1)^nL_{m-n}(x)\,</math>
:<math>F_{n+1}(x)F_{n-1}(x)- F_n(x)^2=(-1)^n\,</math>
:<math>F_{2n}(x)=F_n(x)L_n(x).\,</math>
Closed form expressions, similar to Binet's formula are:<ref name=Springer/>
:<math>F_n(x)=\frac{\alpha(x)^n-\beta(x)^n}{\alpha(x)-\beta(x)},\,L_n(x)=\alpha(x)^n+\beta(x)^n,</math>
where 
:<math>\alpha(x)=\frac{x+\sqrt{x^2+4}}{2},\,\beta(x)=\frac{x-\sqrt{x^2+4}}{2}</math>
are the solutions (in ''t'') of
:<math>t^2-xt-1=0.\,</math>
For Lucas Polynomials ''n'' > 0, we have
:<math>L_n(x)=\sum_{k=0}^{\lfloor n/2\rfloor} \frac{n}{n-k} \binom{n-k}{k} x^{n-2k}.</math>

A relationship between the Fibonacci polynomials and the standard basis polynomials is given by<ref>A proof starts from page 5 in [https://web.archive.org/web/20170202051159/http://cmimc.org/Documents/Archive/AlgebraSolutions_2016.pdf Algebra Solutions Packet (no author)].</ref>
:<math>x^n=F_{n+1}(x)+\sum_{k=1}^{\lfloor n/2\rfloor}(-1)^k\left[\binom nk-\binom n{k-1}\right]F_{n+1-2k}(x).</math>
For example,
:<math>x^4 = F_5(x)-3F_3(x)+2F_1(x)\,</math>
:<math>x^5 = F_6(x)-4F_4(x)+5F_2(x)\,</math>
:<math>x^6 = F_7(x)-5F_5(x)+9F_3(x)-5F_1(x)\,</math>
:<math>x^7 = F_8(x)-6F_6(x)+14F_4(x)-14F_2(x)\,</math>