I'm currently exploring Kapteyn-like series in my research, particularly those of the form:
$$\sum_{n=-\infty}^{\infty} c_n J_{n}(x)$$, where:
- $$c_n = \begin{cases} \pi, & \text{for } n = 0 \\ \frac{j}{n}, & \text{for } n > 0 \\ -\frac{j}{n}, & \text{for } n < 0 \end{cases}$$
Another series with coefficients:
- $$c_n = \begin{cases} \frac{\pi^2}{3}, & \text{for } n = 0 \\ \frac{2(-1)^n}{n^2}, & \text{for } n \neq 0 \end{cases}$$
These series are integral to my work, and I'm seeking insights and references on their applications and properties within information theory and communication engineering.
Your expertise and guidance would be immensely valuable. Thank you for any assistance you can provide.