Dickson polynomial
In mathematics, the Dickson polynomials, denoted Dn(x,α), form a polynomial sequence introduced by L. E. Dickson (1897). They were rediscovered by Brewer (1961) 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 α, they give many examples of permutation polynomials; polynomials acting as permutations of finite fields.
For integer n > 0 and in a commutative ring R with identity (often chosen to be the finite field Fq = GF(q)) the Dickson polynomials (of the first kind) over R are given by
The first few Dickson polynomials are
They may also be generated by the recurrence relation for n ≥ 2,
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