Dickson polynomials

In mathematics, the Dickson polynomials (or Brewer polynomials), denoted D_n(x,α), form a polynomial sequence introduced by L. E. Dickson (1897) and rediscovered by Brewer (1961) in his study of Brewer sums.

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 mainly studied over finite fields, when they are not equivalent to Chebyshev polynomials. One of the main reasons for interest in them is that for fixed α, they give many examples of permutation polynomials: polynomials acting as permutations of finite fields.

D₀(x,α) = 2, and for n > 0 Dickson polynomials (of the first kind) are given by

The first few Dickson polynomials are

The Dickson polynomials of the second kind E_n 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

The D_n satisfy the identities

For n≥2 the Dickson polynomials satisfy the recurrence relation

The Dickson polynomial D_n = y is a solution of the ordinary differential equation

and the Dickson polynomial E_n = y is a solution of the differential equation

Their ordinary generating functions are

Crucially, the Dickson polynomial D_n(x,a) can be defined over rings in which a is not a square, and over rings of characteristic 2; in these cases, D_n(x,a) is often not related to a Chebyshev polynomial.

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 D_n(x,α) (considered as a function of x with α fixed) is a permutation polynomial for the field with q elements if and only if n is coprime to q²−1.

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