Infinite sum of Bessel Functions

Solution 1:

Neumann's addition theorem is given by \begin{align} J_{0}\left(\sqrt{x^{2} + y^{2} - 2 x y \cos\phi}\ \right) = J_{0}(x) J_{0}(y) + 2 \sum_{n=1}^{\infty} J_{n}(x) J_{n}(y) \cos(n\phi). \end{align} Let $\phi = \pi/2$ to obtain \begin{align} J_{0}\left(\sqrt{x^{2} + y^{2}}\ \right) = J_{0}(x) J_{0}(y) + 2 \sum_{n=1}^{\infty} J_{n}(x) J_{n}(y) \cos\left(\frac{n\pi}{2}\right) \end{align} which leads to \begin{align} \sum_{n=1}^{\infty} (-1)^{n} J_{2n}(x) J_{2n}(y) = \frac{1}{2} \left[ J_{0}\left(\sqrt{x^{2} + y^{2}}\ \right) - J_{0}(x) J_{0}(y) \right]. \end{align}