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

Abstract

The Bessel functions of the first kind, Jv(z), with 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/Jv(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.