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 α 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[1] The first few Dickson polynomials are They may also be generated by the recurrence relation for n ≥ 2, 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.
[6][7] The Dn are the unique monic polynomials satisfying the functional equation where α ∈ Fq and u ≠ 0 ∈ Fq2.
The Dickson polynomial y = Dn is a solution of the ordinary differential equation and the Dickson polynomial y = En is a solution of the differential equation Their ordinary generating functions are By the recurrence relation above, Dickson polynomials are Lucas sequences.
The Dickson polynomial Dn(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.
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.