Associated Legendre polynomials

Canonical solutions of the general Legendre equation

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...