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Associated Legendre polynomials

Canonical solutions of the general Legendre equationIn mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre differential equation(1−x2)d2dx2Pℓm(x)−2xddxPℓm(x)+[ℓ(ℓ+1)−m21−x2]Pℓm(x)=0,{\displaystyle \left(1-x^{2} ight){\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}}} ight]P_{\ell }^{m}(x)=0,}or equivalently ddx[(1−x2)ddxPℓm(x)]+[ℓ(ℓ+1)−m21−x2]Pℓm(x)=0,{\displaystyle {\frac {d}{dx}}\left[\left(1-x^{2} ight){\frac {d}{dx}}P_{\ell }^{m}(x) ight]+\left[\ell (\ell +1)-{\frac {m^{2}}{1-x^{2}}} ight]P_{\ell }^{m}(x)=0,}where the indices ℓ and m (which are integers) are referred to as the order and degree of the associated Legendre polynomial respectively.

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