I am currently struggeling with a series(actually two). The problem is, that I can do nothing with them, since this expression is so ugly. I would love to hear about any kind of approximations that you could think of in order to have a more convenient form. $(cos(\alpha))$ is a constant and the actual variables are $r$ and $\theta$:
$$ f_1(r,\theta):=\sum_{l=0}^{\infty}\frac{1}{2l+1}\left(P_{l+1}(\cos(\alpha))-P_{l-1}(\cos(\alpha))\right)\frac{r^l}{R^{l+1}} P_l(cos(\theta)),$$ where $P_l$ is the $l^{\textrm{th}}$ Legendre polynomial.
$$ f_2(r,\theta):=\sum_{l=0}^{\infty}\frac{1}{2l+1}\left(P_{l+1}(\cos(\alpha))-P_{l-1}(\cos(\alpha))\right)\frac{R^l}{r^{l+1}} P_l(cos(\theta)).$$
Any kind of suggest would be helpful. I do not expect to get a linear equation, but maybe one could write a few things differently, approximate something, do not consider a few terms. Just think about things you would do in my situation.
Actually, I can add another one:
$$ f_3(r,\theta):=\sum_{l=0}^{\infty}\frac{1}{2l+1}\left(P_{l+1}(\cos(\alpha))-P_{l-1}(\cos(\alpha))\right) P_l(cos(\theta)).$$