I want to show this equality involving the Bessel function $$x^2=2\sum_{i=1}\frac {(ξ_{0i}^2-4)}{ξ_{0i}^2J_1(ξ_{0i})}J_0(ξ_{0i}x), 0\leq x< 1$$ if $ξ_{0i}$ are the roots of the equation $J_0(x)=0$. What I have done: $f(x)=x^2, f(x)=\sum_{i=1}^\infty c_i J_0(ξ_{0i})$ I try to calculate $c_i$:
$c_i=\frac{2}{J_1^2(ξ_{0i})} \int_0^1x^3J_0(ξ_{0ix})dx$.
After integration by parts and some ectra steps I get this: $c_i=\frac{2}{J_1^2(ξ_{0i})}[ξ_{0i}J_1(ξ_{0i})-2ξ_{0i}^2J_2(ξ_{0i})]$ , but I couldn't get any further... Any help would be appreciated.