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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!

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

David T
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    The formula $J_0(x + y) = J_0(x)J_0(y) - 2J_1(x)J_1(y) + 2J_2(x)J_2(y) - ...$ may come in useful here. – MattyZ May 23 '14 at 01:50

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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}

Ali Caglayan
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Leucippus
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    Drat, I just started work on the same answer! +1 – MattyZ May 23 '14 at 03:37
  • Fantastic, that's just what I needed. I've been coming across a few sums like this one when using Jacobi-Anger expansion on integrands like cos^2(cos(x))*sin^2(sin(x)). Do you have any good resources for more identities or theorems like the one you've given? – David T May 23 '14 at 07:04
  • The first place to often start looking is the bible of Bessel functions, namely, Watson's book. https://archive.org/details/ATreatiseOnTheTheoryOfBesselFunctions A printed version isn't too expensive and is worth it if dealing with Bessel functions of all kinds. – Leucippus May 23 '14 at 14:35