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Let $f=ax+b{x}^q+x^{2q-1}{\in }_q\left[x\right]$. We find explicit conditions on a and b that are necessary and sufficient for f to be a permutation polynomial of ${}_{q²}$. This result allows us to solve a related problem: Let $g_{n,q}{\in }_p\left[x\right]$ (n ≥ 0, $p=cha{r}_q$) be the polynomial defined by the functional equation ${\sum }_{c{\in }_q}{(x+c)}^n=g_{n,q}(x^q-x)$. We determine all n of the form $n=q^{\alpha }-q^{\beta }-1$, α > β ≥ 0, for which $g_{n,q}$ is a permutation polynomial of ${}_{q²}$.

Let q > 2 be a prime power and $f=-x+t{x}^q+x^{2q-1}$, where $t\in {*}_q$. We prove that f is a permutation polynomial of ${}_{q²}$ if and only if one of the following occurs: (i) q is even and $T{r}_{q/2}(1/t)=0$; (ii) q ≡ 1 (mod 8) and t² = -2.