Application of Krein’s series to calculation of sums containing zeros of the Bessel functions

The Bessel functions of the first kind, $J_{\mathrm{v}}(z)$, with $\mathrm{v}>-1$ are considered. On the basis of the general theorem on the representation of the reciprocal of an entire function in the form of Krein’s series, an expansion of the function $1/J_{\mathrm{v}}(z)$ in simple fractions is obtained. This result is used to calculate the sums of series of a certain structure that contain powers of positive zeros of Bessel functions.