What is the closed form of $\sum _{n=1}^{\infty }{\frac { {{\rm J_{0}}\left(2\,n\right)} ^{2}}{{n}^{2}}}$?

Using Maple I am obtaining

$$\sum _{n=1}^{\infty }{\frac { {{\rm J_0}\left(2\,n\right)} ^{2}}{{n}^{2}}} = 0.09845497463$$

• That's not a closed form, it's a numerical approximation. Sep 18, 2014 at 23:35
• You are right Professor Israel, I do not know what is the closed form. Sep 18, 2014 at 23:50
• Do you have any reason to believe that there would be a closed form? Where did you come across this sum? Sep 19, 2014 at 0:01
• You are right, I am not sure if a closed form exists. Sep 19, 2014 at 0:10

Yes, there is a closed form. Expand $\arcsin(t) \chi_{[-1,1]}(t)$ as a Fourier series on $[-\pi/2, \pi/2]$ (where $\chi_{[-1,1]}(t)$ is the indicator function of the interval $[-1,1]$.

$$\arcsin(t) \chi_{[-1,1]} = \sum_{n=1}^\infty \dfrac{J_0(2n) - \cos(2n)}{n} \sin(2nt)$$

Now it seems that $$\sum_{n=1}^\infty \dfrac{\cos(2n)}{n} \sin(2nt) = \cases{ -t - \pi/2 & if -\pi/2 < t < -1\cr -t & if -1 < t < 1\cr -t + \pi/2 & if 1 < t < \pi/2\cr }$$

If $$g(t) = \cases{-t - \pi/2 & if -\pi/2 < t < -1\cr -t + \arcsin(t) & if -1 < t < 1\cr -t + \pi/2 & if 1 < t < \pi/2\cr }$$ Parseval's theorem says

\eqalign{\sum_{n=1}^\infty \dfrac{J_0(2n)^2}{n^2} &=\dfrac{2}{\pi} \int_{-\pi/2}^{\pi/2} g(t)^2\; dt\cr &= \dfrac{\pi^2}{6} + 1 - \dfrac{8}{\pi}}

EDIT:

Somewhat more generally, if $0 \le s \le \pi/2$ we have

$$\arcsin(t/s) \chi_{[-s,s]}(t) = \sum_{n=1}^\infty \dfrac{J_0(2ns) - \cos(2ns)}{n} \sin(2\pi t)$$

and $$\sum_{n=1}^\infty \dfrac{\cos(2ns)}{n} \sin(2nt) = \cases{-t - \pi/2 & if -\pi/2 < t < -s\cr -t & if -s < t < s\cr -t + \pi/2 & if s < t < \pi/2\cr}$$ so that Parseval says $$\sum_{n=1}^\infty \dfrac{J_0(2ns)^2}{n^2} = \dfrac{\pi^2}{6} - 8 \dfrac{s}{\pi} + s^2$$

• Fantastic! I stand corrected in my presumption. Sep 19, 2014 at 6:46
• Very nice computation Professor Israel, congratulations and many thanks. Sep 19, 2014 at 10:44