Two classes of permutation polynomials over finite fields

Two new classes of permutation polynomials over finite fields are presented: (i) f ( x ) = ( 1 − x − x 2 ) x 3 e + 1 2 − 1 − x + x 2 over F 3 e where e is a positive even integer; (ii) g n , p ( x ) = ∑ n p ⩽ l ⩽ n p − 1 n l ( l n − l ( p − 1 ) ) × x n − l ( p − 1 ) over F p e where e is a positive integer such that e ≡ 0 ( mod   2 ) if p = 2 , and n = ( p − 1 ) p m + p 0 e + p 1 e + ⋯ + p ( p − 1 ) e , ( m − 1 , e ) = 1 . The permutation polynomial in (i) answers an open question about reversed Dickson polynomials.