Legendre Polynomials
Definition
Legendre polynomials are defined in various ways.
As Solutions to a Differential Equation
The solutions to the following Legendre differential equation are called Legendre polynomials.
$$ (1-x^{2}) \dfrac{d^{2} y}{dx^{2}} -2x\dfrac{dy}{dx} + l(l+1) y = 0 $$
Rodrigues’ Formula
The following function $P_{l}$ is called a Legendre polynomial.
$$ P_{l}(x) = \dfrac{1}{2^{l} l!} \dfrac{d^{l}}{dx^{l}}(x^{2}-1)^{l} $$
This is known as Rodrigues’ formula.
Explanation
By definition, $P_{n}$ is technically a polynomial ‘function’, but conventionally it is referred to as a Legendre ‘polynomial’. This is the case not only in Korean, but also in English, where it’s not called a polynomial function but a Legendre polynomial.
Legendre polynomials are used in various fields such as mathematics, physics, engineering, etc., because they have many good mathematical properties including orthogonality, and they appear as solutions to the Laplace equation in spherical coordinates. The first few Legendre polynomials are as follows.
$$ \begin{align*} P_{0}(x) &= 1 \\ P_{1}(x) &= x \\ P_{2}(x) &= \dfrac{1}{2}(3x^2-1) \\ P_{3}(x) &= \dfrac{1}{2}(5x^3-3x) \\ P_{4}(x) &= \dfrac{1}{8}(35x^4-30x^2+3) \\ \vdots \end{align*} $$
Properties
Orthogonality
Over the interval $[-1,1]$, Legendre polynomials form an orthogonal set. (Link)
$$ \int_{-1}^{1} P_{l}(x)P_{m}(x) dx =\frac{2}{2l+1}\delta_{lm} $$
Also, Legendre polynomials are orthogonal to any polynomials of lower degree. Let $f(x)$ be any polynomial of a degree lower than $l$. Then,
$$ \int_{-1}^{1}P_{l}(x)f(x)dx=0 $$
Recurrence Relation
Legendre polynomials satisfy the following recurrence formulas. (Link)
$$ (2l+1)P_{l}(x)=P^{\prime}_{l+1}(x)-P^{\prime}_{l-1}(x) $$
$$ lP_{l}(x)=(2l-1)xP_{l-1}(x)-(l-1)P_{l-2}(x) $$
$$ xP^{\prime}_{l}(x)-P^{\prime}_{l-1}(x)=lP_{l}(x) $$
Generating Function
The generating function of Legendre polynomials is as follows. (Link)
$$ \Phi (x,h)=\frac{1}{\sqrt{1-2xh+h^{2}}},\quad |h|<1 $$
By definition, the generating function satisfies the following equation.
$$ \Phi (x,h)=P_{0}(x)+hP_{1}(x)+h^{2}P_{2}(x)+\cdots =\sum \limits_{l=0}^{\infty}h^{l}P_{l}(x) $$