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.
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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 .
 proved that any integral polynomial that is a permutation polynomial for infinitely many prime fields is a composition of Dickson polynomials and linear polynomials (with rational coefficients). This assertion has become known as Schur's conjecture, although in fact Schur did not make this conjecture. Since Fried's paper contained numerous errors, a corrected account was given by , and subsequently gave a simpler proof along the lines of an argument due to Schur.
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
 and , showing that this definition unifies and generalizes the original polynomials of Dickson.
The significant properties of the Dickson polynomials also generalize: