The unnormalized associated Legendre Function (ALF) $P_\ell^m(x)$, $0\leq m\leq \ell$, are defined as $ P_\ell^m(x) = (-1)^m(1-x^2)^{m/2}\frac{d^m}{dx^m}P_\ell(x). $ Here, $P_\ell(x)$ is the [[Legendre Polynomials| Legendre Polynomial]] of degree $\ell$. $m$ is called the order of the associated Legendre function. The factor $(-1)^m$ is called the *Condon-Shortley* phase factor. ALFs can be defined for negative orders through $ P_\ell^{-m} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!} P_\ell^m(x),~0\leq m\leq \ell. $ Note that the factor $(-1)^m$ here is always present and not the same as the Condon-Shortley phase. ## References - Alken, P. _Implementation of Associated Legendre Functions in  GSL_. [https://www.gnu.org/software/gsl/tr/tr001.pdf](https://www.gnu.org/software/gsl/tr/tr001.pdf)