On a conjecture about a class of permutation trinomials

We prove a conjecture by Tu, Zeng, Li, and Helleseth concerning trinomials fα,β(x)=x+αxq(q−1)+1+βx2(q−1)+1∈Fq2[x], αβ≠0, q even, characterizing all the pairs (α,β)∈Fq22 for which fα,β(x) is a permutation of Fq2.