I wonder if the is a way to get a compact form of the sum, $\sum_k i^k J_k(-iz)J_{n-k}(z)$ where $J_k$ and $J_{n-k}$ are Bessel functions.

In particular I'm trying to derive something like the Neumann addition formula,

$\mathscr{C}_\nu (u\pm v) = \sum_{k=-\infty}^{+\infty} \mathscr{C}_{\nu \mp k}(u)J_k(v)$

where $\mathscr{C_{\nu}}$ stands for Bessel functions of the first, second and third type, i.e. $J_{\nu}, Y_{\nu}, H^{(1)}_{\nu}$ and $H^{(2)}_{\nu}$. https://dlmf.nist.gov/10.23

There is also a condition $|v|<|u|$ in the given formula which I don't understand.


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