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:

$P_n^m(x)=(1-x^2{)}^{m/2}\frac{d^m}{d{x}^m}{P}_n(x)$

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.