LegendrePolynomial

The Legendre polynomial $P_n(x)$, orthogonal on $[-1, 1]$ with unit weight and normalised by $P_n(1) = 1$. Provided by @enumeratio/analytic.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaLegendre polynomials↗MathWorldLegendrePolynomial↗DLMF18.3.T1.t1.r10↗FungrimLegendrePolynomial✓↗Fungrim entryWolfram LanguageLegendreP✓↗

LegendrePolynomial(n, x)the degree-$n$ Legendre polynomial at $x$, for an integer $n$.

SageMathlegendre_P(n, x)SymPylegendre(n, x)mpmathlegendre(n, x)Juliaenumeratio_ball((n, x) -> hypergeometric_2f1(-n, n + 1, CC(1), (1 - x) / 2), n, x)

Typed signatures

Type	Package	Applies
(integer, number) -> number	analytic	always

Domain: Special functions

Details

• $P_0 = 1$, $P_1 = x$, and Bonnet's recurrence $(n+1)\,P_{n+1} = (2n+1)\,x\,P_n - n\,P_{n-1}$. The coefficients are rationals whose denominators are powers of 2.
• $\int_{-1}^{1} P_m(x)\,P_n(x)\,dx = \dfrac{2}{2n+1}\,\delta_{mn}$.
• $P_n(1) = 1$ and $P_n(-1) = (-1)^n$. Parity follows the degree: $P_n(-x) = (-1)^n P_n(x)$.
• A negative order folds as Wolfram does: $P_{-n} = P_{n-1}$.
• At an integer $n$ and a symbolic or exact $x$ the result is the expanded polynomial. At a floating-point or complex $x$ the recurrence runs directly on the number.
• The name is Fungrim's. Wolfram calls it LegendreP, which also takes a non-integer degree and, with three arguments, the associated functions; only the integer-degree polynomial is implemented here.

Plot

Examples

7 more kept as data, checked against the oracles

Basic #

$P_2(x) = \tfrac12(3x^2 - 1)$

live ↗

In$\mathrm{LegendrePolynomial}(2, x)$Out$\frac{3x^2}{2}-\frac{1}{2}$

$P_3(x) = \tfrac12(5x^3 - 3x)$

live ↗

In$\mathrm{LegendrePolynomial}(3, x)$Out$\frac{5x^3}{2}-\frac{3x}{2}$

$P_4(x) = \tfrac18(35x^4 - 30x^2 + 3)$

live ↗

In$\mathrm{LegendrePolynomial}(4, x)$Out$\frac{35x^4}{8}-\frac{15x^2}{4}+\frac{3}{8}$

At a floating-point $x$

live ↗

In$\mathrm{LegendrePolynomial}(5, 0.7)$Out$-0.365\,198\,749\,999\,999\,9$

Scope #

At a rational $x$ the value is an exact rational

live ↗

In$\mathrm{LegendrePolynomial}(3, \frac{1}{2})$Out$-\frac{7}{16}$

At a Gaussian integer

live ↗

In$\mathrm{LegendrePolynomial}(3, 2+3i)$Out$-118+18i$

A symbolic order stays unevaluated

live ↗

In$\mathrm{LegendrePolynomial}(n, x)$Out$\mathrm{LegendrePolynomial}(n, x)$

Properties #

$P_n(1) = 1$

live ↗

In$\mathrm{LegendrePolynomial}(6, 1)$Out$1$

$P_n(-1) = (-1)^n$

live ↗

In$\mathrm{LegendrePolynomial}(5, -1)$Out$-1$

$P_{2m}(0) = (-1)^m \binom{2m}{m}/4^m$, here $\binom42/16$

live ↗

In$\mathrm{LegendrePolynomial}(4, 0)$Out$\frac{3}{8}$

Bonnet's recurrence $(n+1)P_{n+1} = (2n+1)xP_n - nP_{n-1}$, at $n = 3$, $x = 3$

live ↗

In$4\mathrm{LegendrePolynomial}(4, 3)=7×3\mathrm{LegendrePolynomial}(3, 3)-3\mathrm{LegendrePolynomial}(2, 3)$Out$\mathrm{True}$

A negative order folds: $P_{-n} = P_{n-1}$

live ↗

In$\mathrm{LegendrePolynomial}(-3, x)$Out$\frac{3x^2}{2}-\frac{1}{2}$

About

Examples	19
Checked against	Wolfram Language 19 agreeSageMath 19 agreeSymPy 19 agreempmath 14 agreeJulia 14 agree
Crosswalk	WikipediaMathWorldDLMFFungrimFungrim entry ×23Wolfram LanguageSageMathSymPympmathJulia

Implementation

primitive · kernela kernel — an algorithm over mutable state that would not be readable as a tree

nativeenginetypescriptpackages/symbols/analysis/analytic/src/legendre.ts

mappedexternalwolfram / mpmath / sympypackages/oracle/src/mappings.ts

LegendreP[n, x]; mpmath.legendre(n, x); sympy.legendre(n, x).

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

See also: ChebyshevT, ChebyshevU