Associated Legendre polynomials

In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre differential equation ( 1 − x 2 ) d 2 d x 2 P ℓ m ( x ) − 2 x d d x P ℓ m ( x ) + [ ℓ ( ℓ + 1 ) − m 2 1 − x 2 ] P ℓ m ( x ) = 0 , {\displaystyle \left(1-x^{2}\right){\frac {d^{2}}{dx^{2}}}P_{\ell }^{m}(x)-2x{\frac {d}{dx}}P_{\ell }^{m}(x)+\left[\ell (\ell +1)-{\frac {m^{2}}{1-x^{2}}}\right]P_{\ell }^{m}(x)=0,} or equivalently d d x [ ( 1 − x 2 ) d d x P ℓ m ( x ) ] + [ ℓ ( ℓ + 1 ) − m 2 1 − x 2 ] P ℓ m ( x ) = 0 , {\displaystyle {\frac {d}{dx}}\left[\left(1-x^{2}\right){\frac {d}{dx}}P_{\ell }^{m}(x)\right]+\left[\ell (\ell +1)-{\frac {m^{2}}{1-x^{2}}}\right]P_{\ell }^{m}(x)=0,} where the indices ℓ and m (which are integers) are referred to as the degree and order of the associated Legendre polynomial respectively. This equation has nonzero solutions that are nonsingular on [−1, 1] only if ℓ and m are integers with 0 ≤ m ≤ ℓ, or with trivially equivalent negative values.

Source: Wikipedia — Associated Legendre polynomials (CC BY-SA 4.0)