We give new descriptions of the factors of Dickson polynomials over finite fields in terms of cyclotomic factors. To do this generalized reciprocal polynomials are introduced and characterized. We also study the factorization of generalized cyclotomic polynomials and their relationship to the factorization of Dickson polynomials.
Fitzgerald, Robert W. and Yucas, Joseph L. "Generalized Reciprocals, Factors of Dickson Polynomials and Generalized Cyclotomic Polynomials over Finite Fields." (Jan 2007).