Dickson polynomial

In mathematics, the Dickson polynomials, denoted D_n(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.

Definition

First kind

For integer n > 0 and α in a commutative ring R with identity (often chosen to be the finite field F_q = GF(q)) the Dickson polynomials (of the first kind) over R are given by^[1]

${\displaystyle D_{n}(x,\alpha )=\sum _{i=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }{\frac {n}{n-i}}{\binom {n-i}{i}}(-\alpha )^{i}x^{n-2i}\,.}$

The first few Dickson polynomials are

${\displaystyle {\begin{aligned}D_{1}(x,\alpha )&=x\\D_{2}(x,\alpha )&=x^{2}-2\alpha \\D_{3}(x,\alpha )&=x^{3}-3x\alpha \\D_{4}(x,\alpha )&=x^{4}-4x^{2}\alpha +2\alpha ^{2}\\D_{5}(x,\alpha )&=x^{5}-5x^{3}\alpha +5x\alpha ^{2}\,.\end{aligned}}}$

They may also be generated by the recurrence relation for n ≥ 2,

${\displaystyle D_{n}(x,\alpha )=xD_{n-1}(x,\alpha )-\alpha D_{n-2}(x,\alpha )\,,}$

with the initial conditions D₀(x,α) = 2 and D₁(x,α) = x.

The coefficients are given at several places in the OEIS^[2][3][4][5] with minute differences for the first two terms.

Second kind

The Dickson polynomials of the second kind, E_n(x,α), are defined by

${\displaystyle E_{n}(x,\alpha )=\sum _{i=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }{\binom {n-i}{i}}(-\alpha )^{i}x^{n-2i}.}$

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

${\displaystyle {\begin{aligned}E_{0}(x,\alpha )&=1\\E_{1}(x,\alpha )&=x\\E_{2}(x,\alpha )&=x^{2}-\alpha \\E_{3}(x,\alpha )&=x^{3}-2x\alpha \\E_{4}(x,\alpha )&=x^{4}-3x^{2}\alpha +\alpha ^{2}\,.\end{aligned}}}$

They may also be generated by the recurrence relation for n ≥ 2,

${\displaystyle E_{n}(x,\alpha )=xE_{n-1}(x,\alpha )-\alpha E_{n-2}(x,\alpha )\,,}$

with the initial conditions E₀(x,α) = 1 and E₁(x,α) = x.

The coefficients are also given in the OEIS.^[6][7]

Properties

The D_n are the unique monic polynomials satisfying the functional equation

${\displaystyle D_{n}\left(u+{\frac {\alpha }{u}},\alpha \right)=u^{n}+\left({\frac {\alpha }{u}}\right)^{n},}$

where α ∈ F_q and u ≠ 0 ∈ F_q².^[8]

They also satisfy a composition rule,^[8]

${\displaystyle D_{mn}(x,\alpha )=D_{m}{\bigl (}D_{n}(x,\alpha ),\alpha ^{n}{\bigr )}\,=D_{n}{\bigl (}D_{m}(x,\alpha ),\alpha ^{m}{\bigr )}\,.}$

The E_n also satisfy a functional equation^[8]

${\displaystyle E_{n}\left(y+{\frac {\alpha }{y}},\alpha \right)={\frac {y^{n+1}-\left({\frac {\alpha }{y}}\right)^{n+1}}{y-{\frac {\alpha }{y}}}}\,,}$

for y ≠ 0, y² ≠ α, with α ∈ F_q and y ∈ F_q².

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

${\displaystyle \left(x^{2}-4\alpha \right)y''+xy'-n^{2}y=0\,,}$

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

${\displaystyle \left(x^{2}-4\alpha \right)y''+3xy'-n(n+2)y=0\,.}$

Their ordinary generating functions are

${\displaystyle {\begin{aligned}\sum _{n}D_{n}(x,\alpha )z^{n}&={\frac {2-xz}{1-xz+\alpha z^{2}}}\\\sum _{n}E_{n}(x,\alpha )z^{n}&={\frac {1}{1-xz+\alpha z^{2}}}\,.\end{aligned}}}$

Links to other polynomials

By the recurrence relation above, Dickson polynomials are Lucas sequences. Specifically, for α = −1, 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.

• The Dickson polynomials with parameter α = 0 give monomials.

${\displaystyle D_{n}(x,0)=x^{n}\,.}$

• The Dickson polynomials with parameter α = 1 are related to Chebyshev polynomials T_n(x) = cos (n arccos x) of the first kind by^[1]

${\displaystyle D_{n}(2x,1)=2T_{n}(x)\,.}$

• Since the Dickson polynomial D_n(x,α) can be defined over rings with additional idempotents, D_n(x,α) is often not related to a Chebyshev polynomial.

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 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.^[9]

Fried (1970) 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 Turnwald (1995), and subsequently Müller (1997) gave a simpler proof along the lines of an argument due to Schur.

Further, Müller (1997) proved that any permutation polynomial over the finite field F_q whose degree is simultaneously coprime to q and less than q^⁠1/4⁠ 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 (k + 1)th kind.^[10] Specifically, for α ≠ 0 ∈ F_q with q = p^e for some prime p and any integers n ≥ 0 and 0 ≤ k < p, the nth Dickson polynomial of the (k + 1)th kind over F_q, denoted by D_n,k(x,α), is defined by^[11]

${\displaystyle D_{0,k}(x,\alpha )=2-k}$

and

${\displaystyle D_{n,k}(x,\alpha )=\sum _{i=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }{\frac {n-ki}{n-i}}{\binom {n-i}{i}}(-\alpha )^{i}x^{n-2i}\,.}$

D_n,0(x,α) = D_n(x,α) and D_n,1(x,α) = E_n(x,α), showing that this definition unifies and generalizes the original polynomials of Dickson.

The significant properties of the Dickson polynomials also generalize:^[12]

• Recurrence relation: For n ≥ 2,

${\displaystyle D_{n,k}(x,\alpha )=xD_{n-1,k}(x,\alpha )-\alpha D_{n-2,k}(x,\alpha )\,,}$

with the initial conditions D_0,k(x,α) = 2 − k and D_1,k(x,α) = x.

• Functional equation:

${\displaystyle D_{n,k}\left(y+\alpha y^{-1},\alpha \right)={\frac {y^{2n}+k\alpha y^{2n-2}+\cdots +k\alpha ^{n-1}y^{2}+\alpha ^{n}}{y^{n}}}={\frac {y^{2n}+{\alpha }^{n}}{y^{n}}}+\left({\frac {k\alpha }{y^{n}}}\right){\frac {y^{2n}-{\alpha }^{n-1}y^{2}}{y^{2}-\alpha }}\,,}$

where y ≠ 0, y² ≠ α.

• Generating function:

${\displaystyle \sum _{n=0}^{\infty }D_{n,k}(x,\alpha )z^{n}={\frac {2-k+(k-1)xz}{1-xz+\alpha z^{2}}}\,.}$