A generalized Lucas sequence and permutation binomials

Abstract

Let p p be an odd prime and q = p m q=p^m . Let l l be an odd positive integer. Let p ≡ − 1 ( mod l ) p\equiv -1~(\textrm {mod}~l) or p ≡ 1 ( mod l ) p\equiv 1~(\textrm {mod}~l) and l ∣ m l\mid m . By employing the integer sequence a n = ∑ t = 1 l − 1 2 ( 2 cos ⁡ π ( 2 t − 1 ) l ) n \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 P(x)=x^r+x^u of the finite field F q \mathbb {F}_q .