 Generating function and norm of Legendre polynomials

With $P_n(1)=1$, the generating function satisfies ${\partial }_x(x^2{\partial }_{x}G)+{\partial }_t((1-t^2){\partial }_{t}G)=0$. Expanding in Legendre polynomials gives $x^2{a}_n^{\prime \prime }+2x{a}_n^{\prime }-n(n+1)a_n=0$. Boundedness at zero removes $x^{-n-1}$, and the value at $t=1$ makes $a_n=x^n$. Orthogonality then gives ${\int }_{-1}^1{G}^{2}dt={\sum }_n{k}_n{x}^{2n}$. 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.