Was calculating something from a physics problem and found myself dealing with the following summation:
$$ \sum\limits_{\text{odd} \text{ } l}^{\infty}P_{l+1}(0)P'_l (\cos{\theta})\sin{\theta}\ $$
By any chance, is there still a way to simplify this further? Is there some trick or identity that could lead to a more neat expression? If not for all values of $\theta$, how about when $\theta = \dfrac{\pi}{2}$?