Special Families of Polynomials

Orthogonal Polynomials

ChebyshevFirst(n) : RngIntElt -> RngUPolElt

ChebyshevT(n) : RngIntElt -> RngUPolElt

Given a positive integer n, this function constructs the Chebyshev polynomial of the first kind T_n(x), where T_n(x) is defined by T_n(x) = cos n θ with x = cos θ.

ChebyshevSecond(n) : RngIntElt -> RngUPolElt

ChebyshevU(n) : RngIntElt -> RngUPolElt

Given a positive integer n, this function constructs the Chebyshev polynomial of the second kind, U_n(x), of degree n - 1. The polynomial is defined by U_n(x) = ((1) /(n)) T_n ' (x) = ((sin n θ) /(sin θ)) where x = cos θ.

LegendrePolynomial(n) : RngIntElt -> RngUPolElt

Given a positive integer n, this function constructs the Legendre polynomial P_n(x) of degree n, where P_n(x) is defined by eqalign(P₀(x) &= 1, P₁(x) = x,
P_n(x) &= (1 /(n)) ((2n - 1) x P_{n - 1}(x) - (n - 1) P_{n - 2}(x)).)

LaguerrePolynomial(n) : RngIntElt -> RngUPolElt

LaguerrePolynomial(n, m) : RngIntElt, RngElt -> RngUPolElt

Given a positive integer n, this function constructs the Laguerre polynomial L_n^m(x) of degree n with parameter m. If m is omitted, it is assumed to be zero if it is not specified. The polynomial satisfies the recurrence relation eqalign(L₀(x) &= 1, L₁(x) = 1 + m - x,
L_n(x) &= (1 /n) (((2n + m - 1) - x) L_{n - 1}^m(x) - (n - 1 + m) L_{n - 2}^m(x)).)

HermitePolynomial(n) : RngIntElt -> RngUPolElt

Given a positive integer n, this function constructs the Hermite polynomial H_n(x) of degree n, where H_n(x) is defined by eqalign(H₀(x) &= 1, H₁(x) = 2x,
H_n(x) &= 2x H_{n - 1}(x) - 2n H_{n - 2}(x).)

GegenbauerPolynomial(n, m) : RngIntElt, RngElt ->RngUPolElt

Given a positive integer n and an integer m, this function constructs the Gegenbauer polynomial C_n^m(x) of degree n with parameter m, where C_n^m(x) is defined by eqalign(C₀^m(x) &= 1, C₁^m(x) = 2 m x,
C_n^m(x) &= (1 /n)(2(n - 1 + m) x C_{n - 1}^m(x) - (n + 2m - 2) C_{n - 2}^m(x)).)

Permutation Polynomials

DicksonFirst(n, a) : RngIntElt, RngElt -> RngUPolElt

Given a positive integer n, this function constructs the Dickson polynomial of the first kind D_n (x, a) of degree n, where D_n (x, a) is defined by D_n(x, a) = ∑_i=0^{⌊n/2 ⌋} (n /(n - i)) ((n - i) choose i) ( - a)ⁱ x^{n - 2i}.

DicksonSecond(n, a) : RngIntElt, RngElt -> RngUPolElt

Given a positive integer n, this function constructs the Dickson polynomial of the second kind E_n (x, a) of degree n, where E_n (x, a) is defined by E_n(x, a) = ∑_i=0^{⌊n/2 ⌋} ((n - i) choose (i)) ( - a)ⁱ x^{n - 2i}.

The Bernoulli Polynomial

BernoulliPolynomial(n) : RngIntElt -> RngUPolElt

Given a positive integer n, this function constructs the n-th Bernoulli polynomial.

Swinnerton-Dyer Polynomials

SwinnertonDyerPolynomial(n) : RngIntElt -> RngUPolElt

Given a positive integer n, this function constructs the n-th Swinnerton-Dyer polynomial, which is defined to be ∏(x ∓ Sqrt(2) ∓ Sqrt(3) ∓ Sqrt(5) ∓ ... ∓ Sqrt(p_n)), where p_i is the i-th prime and the product runs over all 2ⁿ possible combinations of + and - signs. This polynomial lies in Z[x], has degree 2ⁿ, and is irreducible over Z.

See Example H24E7 above which explains more about this class of polynomials, and see also Example H44E2 in the chapter on algebraically closed fields to see how these polynomials are constructed and also for a generalization.