Permutation polynomials with exponents in an arithmetic progression

We examine the permutation properties of the polynomials of the type h(k,r,s)(a) = x(r) (1 + x(s) + ... + x(sk)) over the finite field F-q of characteristic p. We give sufficient and necessary conditions in terms of k and r for h(k,r,1)(x) to be a permutation polynomial over F-q for q = p or p(2). We also prove that if h(k,r,s)(x) is a permutation polynomial over F-q, then (k + 1)(s) = +/-1.