Necessary conditions for reversed Dickson polynomials of the second kind to be permutational

In this paper, we present several necessary conditions for the reversed Dickson polynomial \(E_{n}(1, x)\) of the second kind to be a permutation of \(\mathbb{F}_{q}\). In particular, we give explicit evaluation of the sum \(\sum_{a\in \mathbb{F}_{q}}E_{n}(1,a)\).