I'm wondering how I might go about finding a closed form expression (if it exists) for the following infinite series containing modified Bessel functions:
$\sum_{n=-\infty}^{\infty}cos(n\theta)\ I_n(|x|) \left[K_n(|y|)\ -\ \frac{K_n(a)}{I_n(a)}I_n(|y|)\right]$
where $|x| < |y| < a$ and $cos(\theta) = \frac{x \cdot y}{|x||y|}$. I'm aware of the following addition theorems for (modified) Bessel functions:
$I_0\left(\sqrt{|x|^2 + |y|^2 - 2|x||y|cos(\theta)}\ \right) = \sum_{n=-\infty}^{\infty}cos(n\theta)\ I_n(|x|)\ I_n(|y|)$ $K_0\left(\sqrt{|x|^2 + |y|^2 - 2|x||y|cos(\theta)}\ \right) = \sum_{n=-\infty}^{\infty}cos(n\theta)\ I_n(|x|)\ K_n(|y|)$
so it's really the $\frac{K_n(a)}{I_n(a)}$ multiplicative factor in the second term of the series that makes this non-trivial.