Dickson polynomial
In mathematics, the Dickson polynomials, denoted , form a polynomial sequence introduced by . They were rediscovered by in his study of Brewer sums and have at times, although rarely, been referred to as Brewer polynomials. Over the complex numbers, Dickson polynomials are essentially equivalent to Chebyshev polynomials with a change of variable, and, in fact, Dickson polynomials are sometimes called Chebyshev polynomials. Dickson polynomials are generally studied over finite fields, where they sometimes may not be equivalent to Chebyshev polynomials. One of the main reasons for interest in them is that for fixed , they give many examples of "permutation  polynomials"; polynomials acting as permutations of finite fields. Definition. First kind. For integer and in a commutative ring with identity (often chosen to be the finite field ) the Dickson polynomials (of the first kind) over are given by The first few Dickson polynomials are They may also be generated by the recurrence relation for , with the initial conditions and . Second kind. The Dickson polynomials of the second kind, , are defined by They have not been studied much, and have properties similar to those of Dickson polynomials of the first kind. The first few Dickson polynomials of the second kind are  They may also be generated by the recurrence relation for , with the initial conditions and . Properties. The are the unique monic polynomials satisfying the functional equation where and . They also satisfy a composition rule, The also satisfy a functional equation for , , with and . The Dickson polynomial is a solution of the ordinary differential equation and the Dickson polynomial is a solution of the differential equation Their ordinary generating functions are Links to other polynomials. By the recurrence relation above, Dickson polynomials are Lucas sequences. Specifically, for , the Dickson polynomials of the first kind are Fibonacci polynomials, and Dickson polynomials of  the second kind are Lucas polynomials. By the composition rule above, when α is idempotent, composition of Dickson polynomials of the first kind is commutative. formula_13 formula_14 Permutation polynomials and Dickson polynomials. A permutation polynomial (for a given finite field) is one that acts as a permutation of the elements of the finite field. The Dickson polynomial (considered as a function of with α fixed) is a permutation polynomial for the field with elements if and only if is coprime to . Further, proved that any permutation polynomial over the finite field whose degree is simultaneously coprime to and less than  must be a composition of Dickson polynomials and linear polynomials. Generalization. Dickson polynomials of both kinds over finite fields can be thought of as initial members of a sequence of generalized Dickson polynomials referred to as Dickson polynomials of the th kind. Specifically, for with for some prime and any integers and , the th Dickson polynomial of the th kind over , denoted by , is defined by and The significant properties of the Dickson polynomials also generalize: