Associated Legendre polynomials

Remarks

The factor $(-1)^m$ is the Condon–Shortley phase; some authors leave it out. Examples: $P_\ell^0 = P_\ell$, $P_1^1(x) = -(1 - x^2)^{1/2}$, $P_2^1(x) = -3x (1 - x^2)^{1/2}$ and $P_2^2(x) = 3(1 - x^2)$. Check for $P_2^2$: $P_2(x) = (3x^2 - 1)/2$ has second derivative $3$, so $P_2^2(x) = (1 - x^2) \cdot 3$. Each $P_\ell^m$ solves the associated Legendre equation $(1 - x^2) y'' - 2x y' + \left( \ell(\ell + 1) - \frac{m^2}{1 - x^2} \right) y = 0$ on $(-1, 1)$. For fixed $m$ they are orthogonal: $\int_{-1}^{1} P_k^m(x) P_\ell^m(x) \, dx = \frac{2 (\ell + m)!}{(2\ell + 1)(\ell - m)!} \delta_{k\ell}$. Non-example: the case $m > \ell$ is excluded, because then the $m$-th derivative of $P_\ell$ is zero and the formula gives only the zero function.