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 \(T_n(x) = \cos n \theta\) with \(x = \cos \theta\).

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 \(U_n(x) = {{1} \over {n}} T_n ' (x) = {{\sin n \theta} \over {\sin \theta}}\) where \(x = \cos \theta\).

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

\[\begin{split}\begin{aligned}P_0(x) &= 1, \ P_1(x) = x, \\ P_n(x) &= {1 \over {n}} ((2n - 1) x P_{n - 1}(x) - (n - 1) P_{n - 2}(x)).\end{aligned}\end{split}\]

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

\[\begin{split}\begin{aligned}L_0(x) &= 1, \ L_1(x) = 1 + m - x, \\ L_n(x) &= {1 \over n} (((2n+m-1) - x) L_{n - 1}^m(x) - (n - 1 + m) L_{n - 2}^m(x)).\end{aligned}\end{split}\]

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

\[\begin{split}\begin{aligned}H_0(x) &= 1, \ H_1(x) = 2x, \\ H_n(x) &= 2x H_{n - 1}(x) - 2n H_{n - 2}(x).\end{aligned}\end{split}\]

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

\[\begin{split}\begin{aligned}C_0^m(x) &= 1, \ C_1^m(x) = 2 m x,\\ C_n^m(x) &= {1 \over n}(2(n - 1 + m) x C_{n - 1}^m(x) - (n + 2m - 2) C_{n - 2}^m(x)).\end{aligned}\end{split}\]

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

\[D_n(x, a) = \sum_{i=0}^{\lfloor n/2 \rfloor} {n \over {n - i}} {{n - i} \choose i} (-a)^i x^{n - 2i}.\]

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

\[E_n(x, a) = \sum_{i=0}^{\lfloor n/2 \rfloor} {{n - i} \choose {i}} (-a)^i x^{n - 2i}.\]

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

\[\prod (x \pm \sqrt{2} \pm \sqrt{3} \pm \sqrt{5} \pm \cdots \pm \sqrt{p_n}),\]

where \(p_i\) is the \(i\)-th prime and the product runs over all \(2^n\) possible combinations of \(+\) and \(-\) signs. This polynomial lies in \({\mathbb{Z}}[x]\), has degree \(2^n\), and is irreducible over \({\mathbb{Z}}\).

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