  Let $p$ be an odd prime and $e$ be a positive integer. We completely explain
the permutation binomials and trinomials arising from the reversed Dickson
polynomials of the $(k+1)$-th kind $D_{n,k}(1,x)$ over $\mathbb{F}_{p^e}$ when
$n=p^l+2$, where $l\in \mathbb{N}$.
