I need to simplify this summation: $$\sum_{i=1}^N\sum_{j=1}^N \exp(A^2 \cos(\omega t+\theta_i)\cos(\omega t+\theta_j))$$ where $\theta_i=\frac{2\pi}{N}i$. Without the exponential function the terms could be crossed out in the summation because of symmetry, but I couldn't find a way to simplify the summation when the $\exp(.)$ function is present.
I am also interested to see what happens when $N\to\infty$. Does the summation converge, or does it not converge because the summation fluctuates between different values for odd and even numbers?
