A Note on Permutation Binomials and Trinomials over Finite Fields

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 $l\in \mathbb{N}$.