  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 $D_{n,k}(1,x)$.
