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:

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

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_1^1 (x) =-(1-x^2)^{1/2}}

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

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_2^2 (x) =3(1-x^2)}

etc.