Definition 0.0.1  The associated Legendre polynomials $P_\ell^m(x)$ are a set of polynomials defined on the interval $-1\leq x \leq 1$ for $\ell \in \{0,1,2,3,\dots\}$ and $m \in \{-\ell,-\ell+1,\dots,0,\dots,\ell-1,\ell\}$ by the following expression.
\[P_\ell^m(x) = \frac{(-1)^{m}}{2^\ell\ell!}(1-x^2)^{m/2}\frac{d^{\ell+m}}{dx^{\ell+m}}(x^2-1)^\ell\]