Series of inverse zeros of bessel functions I am interested in the numerical values for the series  $\displaystyle\sum_{n=1}^\infty\frac{1}{j_{0,n}^4}$ and $\displaystyle\sum_{l=1}^\infty\sum_{m=1}^\infty\frac{1}{j_{l,m}^4}$. where  $j_{k,m}$ is the $m-th$ positive zero of the Bessel function of order $k$. Are there known formulas for other exponents (rather than $4$)?
 A: From 'Spectral Sum Rules for the Circular Aharonov-Bohm Quantum Billiard,' F. Steiner, Fortshcr. Phys. 35 1, p 87-114 (1987).  The history of this problem is much older than this reference.
A table is given of the closed forms of the following, for $s=1,2,...10$. I'll stop a short list at s=4.
$$\sum_{k=1}^\infty (j_{a,k})^{-2s}$$
$$s=1: \quad  \frac{1}{4(a+1)}$$
$$s=2: \quad \frac{1}{16(a+1)^2(a+2)}$$
$$s=3: \quad \frac{1}{32(a+1)^3(a+2)(a+3)}$$
$$s=4: \quad \frac{5a+11}{256(a+1)^4(a+2)^2(a+3)(a+4)} $$
A: I do not know any formula for the computation of
$$S_k=\sum _{n=1}^{\infty } \left(j_{0,n}\right){}^{-k}$$ but we can compute them to high accuracy (even if it takes a quite long time).
For $k=4$, the result, for twenty five significant figures is $0.03125000000000000$ and the number of trailing $0$'s is quite impressive (notice that, for $k=2$, the result is $0.25000000000000000$ ). I suppose that this hides something I totally ignore.
I suppose that this hides something I totally ignore. I tried (with no success at all) to use some of the approximations of the zeros.
A: This Rayleigh sums can be calculated using a recurrence formulas, see papers:
[1] Kishore, The Rayleigh function, Proc. Amer. Math. Soc. 14 (1963), 527-533.
[2] Sneddon I. N., On some infinite series involving the zeros of Bessel functions of the first kind, Glasgow Mathematical Journal , Volume 4 , Issue 3 , January 1960 , pp. 144 – 156
[3] Elizalde, E., Leseduarte, S., & Romeo, A. (1993). Sum rules for zeros of Bessel functions and an application to spherical Aharonov-Bohm quantum bags. Journal of Physics A: Mathematical and General, 26(10), 2409–2419. doi:10.1088/0305-4470/26/10/012
[4] J.L. deLyra, On the sums of inverse even powers of zeros of regular Bessel functions, arXiv:1305.0228 [math-ph]
