Associated Legendre polynomials

Summary

The associated Legendre polynomials $P_\ell^m$ are the Legendre functions of integer degree $\ell$ and integer order $m$ on the interval $[-1, 1]$. They are obtained from the Legendre polynomials by differentiating $m$ times and multiplying by $(1 - x^2)^{m/2}$. They give the dependence on the polar angle of the spherical harmonics on the sphere in $\R^3$.