A generalized Lucas sequence and permutation binomials

Let $p$ be an odd prime and $q=p^m$. Let $l$ be an odd positive integer. Let $p\equiv -1~({\rm mod}~l)$ or $p\equiv 1~({\rm mod}~l)$ and $l\mid m$. By employing the integer sequence ${a_n=\sum_{t=1}^{\frac{l-1}{2}} {\left(2\cos{\frac{\pi(2t-1)}{l}}\right)}^n}$, which can be considered as a generalized Lucas sequence, we construct all the permutation binomials $P(x)=x^r+x^u$ of the finite field $\mathbb{F} _q$.