On theorems of Carlitz and Payne on permutation polynomials over finite fields with an application to x−1+L(x)x−1+L(x)

In this paper we give a linearized generalization of a theorem by Carlitz and an odd prime extension of a theorem by Payne concerning permutation polynomials over finite fields. We give an application of our result which classifies the permutation polynomials of the form x−1+L(x)x−1+L(x) in odd characteristic, where L(x)L(x) is a linearized polynomial.