Reversed Dickson polynomials of the $(k+1)$-th kind over finite fields, II

Let $p$ be an odd prime. In this paper, we study the permutation behaviour of the reversed Dickson polynomials of the $(k+1)$-th kind $D_{n,k}(1,x)$ when $n=p^{l_1}+3$, $n=p^{l_1}+p^{l_2}+p^{l_3}$, and $n=p^{l_1}+p^{l_2}+p^{l_3}+p^{l_4}$, where $l_1,l_2$, $l_3$, and $l_4$ are non-negative integers. A generalization to $n=p^{l_1}+p^{l_2}+\cdots +p^{l_i}$ is also shown. We find some conditions under which $D_{n,k}(1,x)$ is not a permutation polynomial over finite fields for certain values of $n$ and $k$. We also present a generalization of a recent result regarding $D_{p^l-1,1}(1,x)$ and present some algebraic and arithmetic properties of $D_{n,k}(1,x)$.