# mathlibdocumentation

ring_theory.polynomial.dickson

# Dickson polynomials #

The (generalised) Dickson polynomials are a family of polynomials indexed by ℕ × ℕ, with coefficients in a commutative ring R depending on an element a∈R. More precisely, the they satisfy the recursion dickson k a (n + 2) = X * (dickson k a n + 1) - a * (dickson k a n) with starting values dickson k a 0 = 3 - k and dickson k a 1 = X. In the literature, dickson k a n is called the n-th Dickson polynomial of the k-th kind associated to the parameter a : R. They are closely related to the Chebyshev polynomials in the case that a=1. When a=0 they are just the family of monomials X ^ n.

## Main definition #

• polynomial.dickson: the generalised Dickson polynomials.

## Main statements #

• polynomial.dickson_one_one_mul, the (m * n)-th Dickson polynomial of the first kind for parameter 1 : R is the composition of the m-th and n-th Dickson polynomials of the first kind for 1 : R.
• polynomial.dickson_one_one_char_p, for a prime number p, the p-th Dickson polynomial of the first kind associated to parameter 1 : R is congruent to X ^ p modulo p.

## TODO #

• Redefine dickson in terms of linear_recurrence.
• Show that dickson 2 1 is equal to the characteristic polynomial of the adjacency matrix of a type A Dynkin diagram.
• Prove that the adjacency matrices of simply laced Dynkin diagrams are precisely the adjacency matrices of simple connected graphs which annihilate dickson 2 1.