Categories: Mathematics.

# Legendre polynomials

The Legendre polynomials are a set of functions that sometimes arise in physics. They are the eigenfunctions $$u(x)$$ of Legendre’s differential equation, which is a (Sturm-Liouville) eigenvalue problem for $$\ell (\ell + 1)$$, where $$\ell$$ turns out to be a non-negative integer:

\begin{aligned} \boxed{ (1 - x^2) u'' - 2 x u' + \ell (\ell + 1) u = 0 } \end{aligned}

The $$\ell$$th-degree Legendre polynomial $$P_\ell(x)$$ is given in the form of a Rodrigues’ formula by:

\begin{aligned} P_\ell(x) &= \frac{1}{2^\ell \ell!} \dv[\ell]{x} (x^2 - 1)^\ell \end{aligned}

The first handful of Legendre polynomials $$P_\ell(x)$$ are therefore as follows:

$\begin{gathered} P_0(x) = 1 \qquad \quad P_1(x) = x \qquad \quad P_2(x) = \frac{1}{2} (3 x^2 - 1) \\ P_3(x) = \frac{1}{2} (5 x^3 - 3 x) \qquad \quad P_4(x) = \frac{1}{8} (35 x^4 - 30 x^2 + 3) \end{gathered}$

And then more $$P_\ell$$ can be computed quickly using Bonnet’s recursion formula:

\begin{aligned} \boxed{ (\ell + 1) P_{\ell + 1}(x) = (2 \ell + 1) x P_\ell(x) - \ell P_{\ell - 1}(x) } \end{aligned}

The derivative of a given $$P_\ell$$ can be calculated recursively using the following relation:

\begin{aligned} \boxed{ \dv{x} P_{\ell + 1} = (\ell + 1) P_\ell(x) + x \dv{x} P_\ell(x) } \end{aligned}

Noteworthy is that the Legendre polynomials are mutually orthogonal for $$x \in [-1, 1]$$:

\begin{aligned} \boxed{ \braket{P_m}{P_n} = \int_{-1}^{1} P_m(x) \: P_n(x) \dd{x} = \frac{2}{2 n + 1} \delta_{nm} } \end{aligned}

As was to be expected from Sturm-Liouville theory. Likewise, they form a complete basis in the Hilbert space of piecewise continuous functions $$f(x)$$ on $$x \in [-1, 1]$$, meaning:

\begin{aligned} \boxed{ f(x) = \sum_{\ell = 0}^\infty a_\ell P_\ell(x) = \sum_{\ell = 0}^\infty \frac{\braket{P_\ell}{f}}{\braket{P_\ell}{P_\ell}} P_\ell(x) } \end{aligned}

Each Legendre polynomial $$P_\ell$$ comes with a set of associated Legendre polynomials $$P_\ell^m(x)$$ of order $$m$$ and degree $$\ell$$. These are the non-singular solutions of the general Legendre equation, where $$m$$ and $$\ell$$ are integers satisfying $$-\ell \le m \le \ell$$:

\begin{aligned} \boxed{ (1 - x^2) u'' - 2 x u' + \Big( \ell (\ell + 1) - \frac{m^2}{1 - x^2} \Big) u = 0 } \end{aligned}

The $$\ell$$th-degree $$m$$th-order associated Legendre polynomial $$P_\ell^m$$ is as follows for $$m \ge 0$$:

\begin{aligned} P_\ell^m(x) = (-1)^m (1 - x^2)^{m/2} \dv[m]{x} P_\ell(x) \end{aligned}

Here, the $$(-1)^m$$ in front is called the Condon-Shortley phase, and is omitted by some authors. For negative orders $$m$$, an additional constant factor is necessary:

\begin{aligned} P_\ell^{-m}(x) = (-1)^m \frac{(\ell - m)!}{(\ell + m)!} P_\ell^m(x) \end{aligned}

Beware, the name is misleading: if $$m$$ is odd, then $$P_\ell^m$$ is actually not a polynomial. Moreover, not all $$P_\ell^m$$ are mutually orthogonal (but some are).