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I am a physicist working on an electrostatic problem and this series popped up: $\sum^{\infty}_{l=0} (P_l(x))^2$ where $P_l$ is the $l$-th Legendre polynomial. Computing this numerically I think the series converges for $x\in(-1,1)$. I don't have the proper knowledge and experience to find out whether it really does. So: 1. Does the series converge? 2. If it does - what is the sum?

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    Although it does not converge, I would think there might be an associated distribution for it, like dirac delta or its derivatives... See http://www-elsa.physik.uni-bonn.de/~dieckman/InfProd/InfProd.html#SeriesxofxLegendrexPolynomials2018-06-15

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This question is delicate, but the series does not always converge. A formula of Laplace ((8.21.2) in Szego's book Orthogonal Polynomials) states that for $0<\theta<\pi$, $ P_n(\cos\theta) = 2^{1/2}(n\pi\sin\theta)^{-1/2}\cos((n+\frac12\theta)-\pi/4) + O(n^{-3/2}).$ Consequently $\sum |P_n(\cos \theta)|^2$ converges if and only if $\sum n^{-1} \cos^2\left((n+\frac12)\theta-\pi/4\right)$ does. But this series does not converge for many values of $\theta$ rationally related to $\pi$, for which there is an arithmetic progression of terms equal to $1/n$. The answer to this question may be relevant when $\theta$ is not rationally related to $\pi$.