Permutational behavior of reversed Dickson polynomials over finite fields II

In this paper, we study the special reversed Dickson polynomial of the form $D_{p^{e_1}+...+p^{e_s}+\ell, k}(1, x)$, where $s, e_1, ..., e_s$ are positive integers, $\ell$ is an integer with $0\le \ell < p$. In fact, by using Hermite criterion we first give an answer to the question that the reversed Dickson polynomials of the forms $D_{p^{s}+1, k}(1, x)$, $D_{p^{s}+2, k}(1, x)$, $D_{p^{s}+3, k}(1, x)$, $D_{p^{s}+4, k}(1, x)$, $D_{p^{s}+p^{t}, k}(1, x)$ and $D_{p^{s}+p^{t}+1, k}(1, x)$ are permutation polynomials of ${\mathbb F}_{q}$ or not. Finally, utilizing the recursive formula of the reversed Dickson polynomials, we represent $D_{p^{e_1}+...+p^{e_s}+\ell, k}(1, x)$ as the linear combination of the elementary symmetric polynomials with the power of $1-4x$ being the variables. From this, we present a necessary and sufficient condition for $D_{p^{e_1}+...+p^{e_s}+\ell, k}(1, x)$ to be a permutation polynomial of ${\mathbb F}_{q}$.