Reversed Dickson Polynomials of the Third Kind

Let $p$ be a prime and $q=p^e$. We discuss the properties of the reversed Dickson polynomial $D_{n,2}(1,x)$ of the third kind. We also give several necessary conditions for the reversed Dickson polynomial of the third kind $D_{n,2}(1,x)$ to be a permutation of $\mathbb{F}_{q}$. In particular, we give explicit evaluation of the sum $\sum_{a\in \mathbb{F}_q}D_{n,2}(1,a)$.