I just started learning Legendre Differential Equation. From what I learn the solutions to it is the Legendre polynomial.(adsbygoogle = window.adsbygoogle || []).push({});

For the legendre DE,what is the l in it?Is it like a variable like y and x, just a different variable instead?

Legendre Differential Equation: $$(1-x^2) \frac{d^2y}{dx^2} - 2x\frac{dy}{dx} + l(l+1)y = 0$$

Butwhat is the differences between the associated Legendre polynomial and unassociated Legendre polynomial?When do we use the associated one or the unassociated one, since both are solutions to the Legendre Differential Equation?

Unassociated Legendre Polynomial: $$P_l(x) = \sum^M_{m=0}(-1)^m\frac{(2l-2m)!}{2^lm!(l-m)!(l-2m)!}x^{l-2m}$$

Associated Legendre Polynomial: $$P^m_l(x) = (-1)^m(1-x^2)^{m/2}\frac{d^m}{dx^m}(P_l(x))$$

Lastly,where did the m come from?

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# I Legendre Differential Equation

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