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.
Contents
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]}
The first few Dickson polynomials are
They may also be generated by the recurrence relation for n ≥ 2,
with the initial conditions D_{0}(x,α) = 2 and D_{1}(x,α) = x.
Second kind
The Dickson polynomials of the second kind, E_{n}(x,α), 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 n ≥ 2,
with the initial conditions E_{0}(x,α) = 1 and E_{1}(x,α) = x.
Properties
The D_{n} are the unique monic polynomials satisfying the functional equation
where α ∈ F_{q} and u ≠ 0 ∈ F_{q2}.^{[2]}
They also satisfy a composition rule,^{[2]}
The E_{n} also satisfy a functional equation^{[2]}
for y ≠ 0, y^{2} ≠ α, with α ∈ F_{q} and y ∈ F_{q2}.
The Dickson polynomial y = D_{n} is a solution of the ordinary differential equation
and the Dickson polynomial y = E_{n} 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 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.
 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]}
 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 q^{2} − 1.^{[3]}
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.^{[4]} 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^{[5]}
and
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:^{[6]}
 Recurrence relation: For n ≥ 2,

 with the initial conditions D_{0,k}(x,α) = 2 − k and D_{1,k}(x,α) = x.
 Functional equation:

 where y ≠ 0, y^{2} ≠ α.
 Generating function:
Notes
 ^ ^{a} ^{b} Lidl & Niederreiter 1983, p. 355
 ^ ^{a} ^{b} ^{c} Mullen & Panario 2013, p. 283
 ^ Lidl & Niederreitter 1983, p. 356
 ^ Wang, Q.; Yucas, J. L. (2012), "Dickson polynomials over finite fields", Finite Fields and their Applications, 18: 814–831
 ^ Mullen & Panario 2013, p. 287
 ^ Mullen & Panario 2013, p. 288
References
 Brewer, B. W. (1961), "On certain character sums", Transactions of the American Mathematical Society, 99: 241–245, doi:10.2307/1993392, ISSN 00029947, JSTOR 1993392, MR 0120202, Zbl 0103.03205
 Dickson, L. E. (1897). "The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group I,II". Ann. of Math. The Annals of Mathematics. 11 (1/6): 65–120; 161–183. doi:10.2307/1967217. ISSN 0003486X. JFM 28.0135.03. JSTOR 1967217.
 Fried, Michael (1970). "On a conjecture of Schur". Michigan Math. J. 17: 41–55. doi:10.1307/mmj/1029000374. ISSN 00262285. MR 0257033. Zbl 0169.37702.
 Lidl, R.; Mullen, G. L.; Turnwald, G. (1993). Dickson polynomials. Pitman Monographs and Surveys in Pure and Applied Mathematics. 65. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York. ISBN 0582091195. MR 1237403. Zbl 0823.11070.
 Lidl, Rudolf; Niederreiter, Harald (1983). Finite fields. Encyclopedia of Mathematics and Its Applications. 20 (1st ed.). AddisonWesley. ISBN 0201135191. Zbl 0866.11069.
 Mullen, Gary L. (2001) [1994], "Dickson polynomials", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104
 Mullen, Gary L.; Panario, Daniel (2013), Handbook of Finite Fields, CRC Press, ISBN 9781439873786
 Müller, Peter (1997). "A Weilbound free proof of Schur's conjecture". Finite Fields and Their Applications. 3: 25–32. doi:10.1006/ffta.1996.0170. Zbl 0904.11040.
 Rassias, Thermistocles M.; Srivastava, H.M.; Yanushauskas, A. (1991). Topics in Polynomials of One and Several Variables and Their Applications: A Legacy of P.L.Chebyshev. World Scientific. pp. 371–395. ISBN 9810206143.
 Turnwald, Gerhard (1995). "On Schur's conjecture". J. Austral. Math. Soc. Ser. A. 58 (03): 312–357. doi:10.1017/S1446788700038349. MR 1329867. Zbl 0834.11052.
 Young, Paul Thomas (2002). "On modified Dickson polynomials" (PDF). Fibonacci Quarterly. 40 (1): 33–40.