Definition and Properties

The associated Legendre polynomials $P_\ell^m(x)$ are the solution to the differential equation

\[\begin{align} (1-x^2) \frac{d^2}{dx^2}P_\ell^m(x) - 2x \frac{d}{dx}P_\ell^m(x) + \left[ \ell(\ell+1) - \frac{m^2}{1-x^2} \right] P_\ell^m(x) = 0 \end{align}\]

which arises as the colatitude $\theta$ part of solving Laplace's equation $\nabla^2 \psi + \lambda\psi = 0$ in spherical coordinates (where $x = \cos(\theta)$).

There are several different conventions used to define $P_\ell^m$ that provide different properties, but the convention used here is typical of quantum mechanics and obeys the following properties:

  • Solutions only exist for integer $\ell$ and $m$, where $\ell ≤ 0$ and $|m| \le \ell$.

  • The associated Legendre functions are normalized such that $P_0^0$ is unity and have orthogonality conditions,

    \[\begin{align} \int_{-1}^1 P_\ell^m(x) P_{\ell'}^{m}(x)\,\mathrm{d}x = \frac{2}{2\ell+1} \frac{(\ell+m)!}{(\ell-m)!} \delta_{\ell\ell'} \end{align}\]

    for constant $m$ and

    \[\begin{align} \int_{-1}^1 \frac{P_\ell^m(x) P_{\ell}^{m'}(x)}{1-x^2}\,\mathrm{d}x = \frac{1}{m} \frac{(\ell+m)!}{(\ell-m)!} \delta_{mm'} \end{align}\]

    for constant $\ell$, where $\delta$ is the Kronecker delta.

  • The phase convention for the Legendre functions is chosen such that the negative orders are related to positive orders according to,

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

  • The Legendre functions can be enumerated for non-negative $m$ using the three following recursion relations (given the initial condition $P_0^0(x)$):

    \[\begin{align} P_{\ell+1}^{\ell+1}(x) &= -(2\ell+1)\sqrt{1-x^2} P_\ell^\ell(x) \label{eqn:std_rr_1term_lm} \\ P_{\ell+1}^\ell(x) &= x(2\ell+1)P_\ell^\ell(x) \label{eqn:std_rr_1term_l} \\ (\ell - m + 1)P_{\ell+1}^m(x) &= (2\ell+1)xP_\ell^m(x) - (\ell+m)P_{\ell-1}^m(x) \label{eqn:std_rr_2term} \end{align}\]