Associated Legendre functions

The associated Legendre functions $P_{n}^{m}(x)$ are polynomials which are most conveniently defined in terms of derivatives of the Legendre polynomials:

$P_{n}^{m}(x)=(1-x^{2})^{m/2}{\frac {d^{m}}{dx^{m}}}P_{n}(x)$

The first associated Legendre polynomials are:

$P_{0}^{0}(x)=1$

$P_{1}^{0}(x)=x$

$P_{1}^{1}(x)=-(1-x^{2})^{1/2}$

$P_{2}^{0}(x)={\frac {1}{2}}(3x^{2}-1)$

$P_{2}^{1}(x)=-3x(1-x^{2})^{1/2}$

$P_{2}^{2}(x)=3(1-x^{2})$

etc.