= Generating function and norm of Legendre polynomials
{title2=$(1-2xt+x^2)^{-1/2}=\sum_{n\geq0}x^nP_n(t)$}
With $P_n(1)=1$, the generating function satisfies $\partial_x(x^2\partial_xG)+\partial_t((1-t^2)\partial_tG)=0$. Expanding in <Legendre polynomials> gives $x^2a_n''+2xa_n'-n(n+1)a_n=0$. Boundedness at zero removes $x^{-n-1}$, and the value at $t=1$ makes $a_n=x^n$. Orthogonality then gives $\int_{-1}^1G^2dt=\sum_nk_nx^{2n}$. The integral is $x^{-1}\log((1+x)/(1-x))$, so $k_n=2/(2n+1)$. Uniform analyticity on compact subdiscs $|x|<1$ justifies the generating expansion and termwise calculations.
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