= Rodrigues formula for Legendre polynomials
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{title2=$P_l(x)=\frac1{2^ll!}\frac{d^l}{dx^l}(x^2-1)^l$}
This instance of <Rodrigues' formula> constructs the <Legendre polynomial> for every nonnegative <integer> $l$. Put $v=(x^2-1)^l$; then $(x^2-1)v'=2lxv$. Differentiating $l+1$ times by the <Leibniz rule> gives $(x^2-1)y''+2xy'-l(l+1)y=0$ for $y=v^{(l)}$, which is the <Legendre differential equation> after reversing signs. The leading term of the normalized derivative is $(2l)!x^l/[2^l(l!)^2]$, proving its <polynomial degree> is exactly $l$. At $x=1$, the product $(x+1)^l(x-1)^l$ contributes only the term with all $l$ derivatives on the second factor, giving $v^{(l)}(1)=2^ll!$. Thus the normalization is $P_l(1)=1$.
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