Specific permutation polynomials over finite fields

We present new classes of permutation polynomials over finite fields. If q is the order of the finite field, some of these polynomials have the form xrf(x(q−1)/d), where d|(q−1). We also present some permutation polynomials involving the trace function, which plays an additive role analogous to x(q−1)/d. Finally, we present a generalization involving other symmetric functions of x,xp,…,xq/p.