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:

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^m_n(x)= (1-x^2)^{m/2} \frac{d^m}{dx^m} P_n(x) }

The first associated Legendre polynomials are:

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_0^0 (x) =1}

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_1^0 (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}}

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

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_2^1 (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.