  We present a non-perturbative, divergence-free series expansion of Green's
functions using effective operators. The method is especially suited for
computing correlators of complex operators as a series of correlation functions
of simpler forms. We apply the method to study low-energy excitations in
resonant inelastic x-ray scattering (RIXS) in doped one- and two-dimensional
single-band Hubbard models. The RIXS operator is expanded into polynomials of
spin, density, and current operators weighted by fundamental x-ray spectral
functions. These operators couple to different polarization channels resulting
in simple selection rules. The incident photon energy dependent coefficients
help to pinpoint main RIXS contributions from different degrees of freedom. We
show in particular that, with parameters pertaining to cuprate superconductors,
local spin excitation dominates the RIXS spectral weight over a wide doping
range in the cross-polarization channel.
