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 ${\sum }_{a\in {\mathbb{F}}_q}{D}_{n,2}(1,a)$.