  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.
