Fibonacci polynomial
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: █
