A generalized Lucas sequence and permutation binomials

Abstract:

Let $p$ be an odd prime and $q=p^m$. Let $l$ be an odd positive integer. Let $p\equiv -1~(\textrm {mod}~l)$ or $p\equiv 1~(\textrm {mod}~l)$ and $l\mid m$. By employing the integer sequence $\displaystyle {a_n=\sum _{t=1}^{\frac {l-1}{2}} {\left (2\cos {\frac {\pi (2t-1)}{l}}\right )}^n}$, 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 $\mathbb {F}_q$.