This paper aims to achieve analytical descriptions of specific classes of finite-energy non-diffracting beams, viz. The so-called Frozen Waves, envisioning applications in optical trapping. Such solutions to the Fresnel diffraction integral can be constructed from specific discrete superpositions of finiteenergy zero-order scalar Bessel-Gauss beams. Here, we present expressions for their beam shape coefficients in the context of the generalized Lorenz-Mie theory. The paraxial regime is valid for all Bessel-Gauss beams, thus allowing the method here presented to be purely analytic. The analyticity avoids both extensive numerical computation and optimization schemes. Radiation pressure cross sections, which are proportional to optical forces, are then evaluated for Rayleigh particles as an example of application. We expect Frozen Waves to serve, in the near future, as alternative laser beams in biomedical optics and in the optical micromanipulation of biological or auxiliary particles.