The study for the permutation of polynomials has wide and important applications in the fields of algebra, combinatorics, number theory, coding theory, cryptography and so on. In this paper, we mainly study Fibonacci polynomials, by calculating the sum of equal powers of their function values, we obtain the necessary conditions for determining these Fibonacci polynomials defined in finite fields as permutation polynomials, solve the open problem raised by Fernando and Rashid. These conditions generalize the existing conclusions on the permutation of Fibonacci polynomials.
