On summation of Kapteyn series

We consider the summation of Kapteyn series of several kinds and obtain some relations among the sums. We paid special attention to cases when sums are transcendental functions. Their asymptotic behaviors are obtained. In some cases the integral representations for sums are found. As an application we find the radiation free lifetime of an electron moving in a constant external field. It is noted that the laboratory lifetime increases with energy for ultrarelativistic electron in ultrastrong external field.