Permutation polynomials (PPs) of the form $(x^{ q} -x + c)^{ \frac{ q^2 -1}{ 3}+1} +x$ over $\mathbb{ F}_{ q^2}$ were presented by Li, Helleseth and Tang [Finite Fields Appl. 22 (2013) 16--23]. More recently, we have constructed PPs of the form $(x^{ q} +bx + c)^{ \frac{ q^2 -1}{ d}+1} -bx$ over $\mathbb{ F}_{ q^2}$, where $d=2, 3, 4, 6$ [Finite Fields Appl. 35 (2015) 215--230]. In this paper we concentrate our efforts on the PPs of more general form \[ f(x)=(ax^{ q} +bx +c)^r \phi((ax^{ q} +bx +c)^{ (q^2 -1)/d}) +ux^{ q} +vx~~\text{ over $\mathbb{ F}_{ q^2}$}, \] where $a,b,c,u,v \in \mathbb{ F}_{ q^2}$, $r \in \mathbb{ Z}^{ +}$, $\phi(x)\in \mathbb{ F}_{ q^2}[x]$ and $d$ is an arbitrary positive divisor of $q^2-1$. The key step is the construction of a commutative diagram with specific properties, which is the basis of the Akbary--Ghioca--Wang (AGW) criterion. By employing the AGW criterion two times, we reduce the problem of determining whether $f(x)$ permutes $\mathbb{ F}_{ q^2}$ to that of verifying whether two more polynomials permute two subsets of $\mathbb{ F}_{ q^2}$. As a consequence, we find a series of simple conditions for $f(x)$ to be a PP of $\mathbb{ F}_{ q^2}$. These results unify and generalize some known classes of PPs.
Categories: math.NT