I am trying to prove the following recurrence relation for Dickson polynomial:
$D_{n+2}(X,a)=XD_{n+1}(X,a)−aD_n(X,a) \ \text{for} \ n≥0 \ \text{with initial conditions} \ D_0(X,a)=2 \ \text{and} \ D_1(X,a)=X$
For $n=2$ recurrence is true, but I have problems with the second step.
EDIT: The definition of Dickson polynomial is: Let $\mathbb{K}$ is field and let $a \in \mathbb{K}$, $n$-th Dickson polynomial $D_n(x,a)$ is \begin{equation} D_n(z+a/z,a)=z^n+(a/z)^n. \end{equation}