Given $-1 \leq x \leq 1$ and $0 \leq \eta \leq 1$,
I am interested in computing
$$
E(x,\eta) = \sum_{\ell = 0}^{+ \infty} |P_{\ell} (0)|^{2} \, P_{\ell} (x) \, \eta^{\ell} ,
$$
with $P_{\ell}$ the usual Legendre polynomials.
Can one find a closed form expression for this function?
NB1: Should this be useful, $E(x,\eta)$ is, in essence, the gravitational potential between two massive circles centered around the same location, inclined by a respective angle $\cos^{-1}(x)$ and with a ratio of radii given by $\eta$.
NB2: Of course, in the limit $x \to 1$ or $\eta \to 0$, I expect that I could perform appropriate limited developments.
Best Answer
As $P_{2n+1}(0)=0$ and $P_{2n}(0)=(-1)^n\frac{(1/2)_n}{n!}$, the proposed series reads \begin{align} E(x,\eta) &= \sum_{\ell = 0}^{+ \infty} |P_{\ell} (0)|^{2} \, P_{\ell} (x) \, \eta^{\ell} \\ &=\sum_{n=0}^\infty \frac{\left(\frac{1}{2}\right)_{n}^{2} }{n !^{2}}P_{2n}(x)\eta^{2n} \end{align}
Interestingly, such a series was discussed in connection with Ramanujan type series for $1/\pi$ in this paper (see also here). Wan and Zudilin prove the following identity \begin{align} \sum_{n=0}^\infty\frac{\left(\frac{1}{2}\right)_{n}^{2} }{n !^{2}}\,&P_{2n} \left( \frac{\left(X +Y \right) \left(1-X Y\right)}{\left(X -Y \right) \left(1+X Y \right)}\right) \left(\frac{X -Y}{X Y +1}\right)^{2 n}\\ &=\frac{\left(1+X Y\right)}{2} \,_2F_1\left(\frac{1}{2},\frac{1}{2};1;1-X^{2}\right)\,_2F_1\left(\frac{1}{2},\frac{1}{2};1;1-Y^{2}\right) \end{align} the domain of validity for the parameters $X,Y$ is relatively complex, but numerical experiments seem to indicate that most of the values of $-1\le x\le 1$ and $0\le \eta\le1$ (if not all) could be covered.