# Bessel equation relation

Define Bessel function as: $$J_a(x)=\sum_{n=0}^{\infty}{{(-1)^n}\over{\Gamma(a+n+1)n!}}\left({{x}\over{2}}\right)^{a+2n}.$$ Where $a$ not an integer.

Need to show $$J_{a+1}(x)J_{-a}(x)+J_a(x)J_{-(a+1)}(x)=-{{2\sin(a\pi)}\over{\pi x}}.$$

Note that the Wronskian of $$W(J_a(x),J_{-a}(x))=-{{2\sin(a\pi)}\over{\pi x}}.$$

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So are you allowed to use that result on Wronskian or you need to prove it also? –  Norbert Jul 14 '13 at 9:39

P.S. The differentiation formulas can be proven either directly using series representations or by verifying that e.g. $\displaystyle J'_a(x)-\frac{a}{x}J_a(x)$ satisfies the same differential equation as $-J_{a+1}(x)$ and comparing their behaviour as $x\rightarrow 0$.