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: █