Mathematica gives a following identity: \begin{equation} (1) J_{1/3}(x) J_{2/3}(x) + J_{-1/3}(x)J_{-2/3}(x) = \frac{\sqrt{3}}{\pi x} \end{equation} How do I prove this identity?
I was trying to use the recurrence relations: \begin{equation} J_{n+1}(x) + J_{n-1}(x) = \frac{2 n}{x} J_n(x) \end{equation} to tackle equation (1) but I do not think this will lead to the right direction since using the recurrence will introduce Bessel functions of higher(lower) orders than the orders currently present in the identity. Can anybody give me a hint as to how to prove that identity?