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I came across the following sum in my work involving the infinite sum of a product of Bessel functions. Does anyone have any idea of how to express this in a simpler form? ‘a’ and ‘b’ are positive numbers, and I am also interested in the case where a=b. Thanks!

$$\sum_{n=1}^{\infty}(-1)^{n}J_{2n}(a)J_{2n}(b)$$

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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}

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