Some permutation pentanomials over finite fields of even characteristic

In a recent paper [30] Zhang et al. constructed 17 families of permutation pentanomials of the form x t + x r 1 (q − 1) + t + x r 2 (q − 1) + t + x r 3 (q − 1) + t + x r 4 (q − 1) + t over F q 2 where q = 2 m . In this paper for 14 of these 17 families we provide a simple explanation as to why they are permutations. We also extend these 14 families into three general classes of permutation pentanomials over F q 2 .