Summation of Reciprocal of Zero of Order 1 Bessel Function by Order 0 Bessel Function of it

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Theorem

$\displaystyle \sum_{n \mathop = 1}^\infty \dfrac 1 {x_n \map {J_0 } {x_n} } = 0 \cdotp 38479 \ldots$

where:

$x_n$ is the $n$th zero of the order $1$ Bessel function of the first kind
$\map {J_0 } {x_n}$ is the order $0$ Bessel function of the first kind of $x_n$.


Historical Note

This result is reported in François Le Lionnais and Jean Brette: Les Nombres Remarquables of $1983$, with no indication of context.

It is not revealed why this result is significant.


Sources