Third Power of the Reversed Dickson Polynomial over Finite Fields

Let $D_n(1,x)$ be the $n$th reversed Dickson polynomial. The power sums ${\sum }_{a\in {\mathbb{F}}_q}{D}_n(1,a{)}^i$, $i=1,2$, have been determined recently. In this paper we give an evaluation of the sum ${\sum }_{a\in {\mathbb{F}}_q}{D}_n(1,a{)}^3$. This result implies new necessary conditions for $D_n(1,x)$ to be a permutation polynomial over ${\mathbb{F}}_q$.