TITLE: Does the series of squares of Legendre polynomials converge?
QUESTION [8 upvotes]: 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?

REPLY [5 votes]: 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
$\pi$.