  We derive an extension to the quantum regression theorem which facilitates
the calculation of two-time correlation functions and emission spectra for
systems undergoing non-Markovian evolution. The derivation exploits projection
operator techniques, with which we obtain explicit equations of motion for the
correlation functions, making only a second order expansion in the
system--environment coupling strength, and invoking the Born approximation at a
fixed initial time. The results are used to investigate a driven semiconductor
quantum dot coupled to an acoustic phonon bath, where we find the non-Markovian
nature of the dynamics has observable signatures in the form of phonon
sidebands in the resonance fluorescence emission spectrum. Furthermore, we use
recently developed non-Markovianity measures to demonstrate an associated flow
of information from the phonon bath back into the quantum dot exciton system.
