Given a positive integer n, this function constructs the Chebyshev polynomial of the first kind Tn(x), where Tn(x) is defined by Tn(x) = cos n θ with x = cos θ.
Given a positive integer n, this function constructs the Chebyshev polynomial of the second kind, Un(x), of degree n - 1. The polynomial is defined by Un(x) = ((1) /(n)) Tn ' (x) = ((sin n θ) /(sin θ)) where x = cos θ.
Given a positive integer n, this function constructs the Legendre polynomial Pn(x) of degree n, where Pn(x) is defined by eqalign(P0(x) &= 1, P1(x) = x,
Pn(x) &= (1 /(n)) ((2n - 1) x Pn - 1(x) - (n - 1) Pn - 2(x)).)
Given a positive integer n, this function constructs the Laguerre polynomial Lnm(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,
Ln(x) &= (1 /n) (((2n + m - 1) - x) Ln - 1m(x) - (n - 1 + m) Ln - 2m(x)).)
Given a positive integer n, this function constructs the Hermite polynomial Hn(x) of degree n, where Hn(x) is defined by eqalign(H0(x) &= 1, H1(x) = 2x,
Hn(x) &= 2x Hn - 1(x) - 2n Hn - 2(x).)
Given a positive integer n and an integer m, this function constructs the Gegenbauer polynomial Cnm(x) of degree n with parameter m, where Cnm(x) is defined by eqalign(C0m(x) &= 1, C1m(x) = 2 m x,
Cnm(x) &= (1 /n)(2(n - 1 + m) x Cn - 1m(x) - (n + 2m - 2) Cn - 2m(x)).)
Given a positive integer n, this function constructs the Dickson polynomial of the first kind Dn (x, a) of degree n, where Dn (x, a) is defined by Dn(x, a) = ∑i=0⌊n/2 ⌋ (n /(n - i)) ((n - i) choose i) ( - a)i xn - 2i.
Given a positive integer n, this function constructs the Dickson polynomial of the second kind En (x, a) of degree n, where En (x, a) is defined by En(x, a) = ∑i=0⌊n/2 ⌋ ((n - i) choose (i)) ( - a)i xn - 2i.
Given a positive integer n, this function constructs the n-th Bernoulli polynomial.
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(pn)), where pi 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 H43E2 in the chapter on algebraically closed fields to see how these polynomials are constructed and also for a generalization.