I am interested in computing a closed-form for the infinite series $$ \sum_{k \geq 1} \frac{(-1)^k}{k^2} \left( I_{k - \frac{1}{2}} \left( z \right) + I_{k + \frac{1}{2}} \left( z \right) \right) \cos{\left( 2 k \theta \right)} , $$ for $\theta \in [0, 2\pi)$, $z > 0$, and where $I_n$ is the modified Bessel function of order $n$. I would also be happy to know closed-forms for the special cases $$ \sum_{k \geq 1} \frac{(-1)^k}{k^2} \left( I_{k - \frac{1}{2}} (x) + I_{k + \frac{1}{2}} (x) \right) , $$ and $$ \sum_{k \geq 1} \frac{1}{k^2} \left( I_{k - \frac{1}{2}} (x) + I_{k + \frac{1}{2}} (x) \right) , $$ which corresponded to $\theta = 0$ and $\theta = \pi / 2$, respectively.
The derivative of the original series with respect to $\theta$ appears related to the Jacobi-Anger relations, but the half integral order prevents a direct adaptation. This question also seems related: Sum of Bessel functions and integral of exponential of sine or cosine. I am thinking the series could be related to $e^{z \cosh{ \theta }}$ or $e^{z \sinh{ \theta }}$ since $I_{k \pm 1/2} (z)$ is are related to $\cosh$ and $\sinh$.
