On a family of permutation polynomials over finite fields

The topic of permutation polynomials over finite fields is important both in theory and in engineering application. Let p be an odd prime number and q a power of p. A necessary and sufficient condition for a family of polynomials over F_q to be permutation polynomials is presented. Moreover, the number of permutation polynomials of this type is obtained. The polynomials discussed in this note are generalizations and improvement of those discussed by Xi-Wang Cao(2009).