Identity for the Bessel function

Question

I'm sure this is easy, but I can't find it yet.

What is $J_n(-x)$? Where $J_n$ is the usual Bessel function of integer order.

In particular, I'm looking to the sum $\sum_{n=-\infty}^\infty J_n(-x) = \sum_{n=-\infty}^\infty f_n(-1)J_n(x) $,

I want to find this $f_n(-1)$.

Answer 1

From definition one can write $[1]$ $$J_n(x)={1\over \pi}\int_0^\pi \cos(n\tau-x\sin\tau)d\tau$$therefore $$J_{-n}(-x){={1\over \pi}\int_0^\pi \cos(-n\tau+x\sin\tau)d\tau\\={1\over \pi}\int_0^\pi \cos(n\tau-x\sin\tau)d\tau\\=J_n(x)}$$also$$J_{-n}(x)=(-1)^nJ_n(x)$$finally$$J_n(-x)=(-1)^nJ_n(x)$$

Answer 2

According to Wikipedia:

$$ \lambda^{-\nu} J_\nu(\lambda z) = \sum_{n=0}^{\infty} \frac{1}{n!}\left( \frac{(1-\lambda^2)z}{2} \right)^n J_{\nu+n}(z) $$

Hence, picking $\lambda = -1$:

$$ (-1)^{-\nu} J_\nu(- z) = J_{\nu}(z) $$