  It is well known that our motion with respect to the cosmic microwave
background (CMB) rest frame introduces a large dipolar CMB anisotropy, with an
amplitude ~beta=v/c~10^{-3}. In addition it should lead to a small breaking of
statistical isotropy which becomes most notable at higher multipoles. In
principle this could be used to determine our velocity with respect to the CMB
rest frame using high angular resolution data from Planck, without directly
relying on the amplitude and direction of the CMB dipole, allowing us to
constrain cosmological models in which the cosmic dipole arises partly from
large-scale isocurvature perturbations instead of being fully motion-induced.
Here we derive simple recursion relations that allow precise computation of the
motion-induced coupling between different spherical harmonic coefficients.
Although the lowest order approximations for the coupling kernel can be
deficient by factors of 2-5 at multipoles l~1000-3000, using our results for
the aberration kernel we explicitly confirm that for a statistical detection of
the aberration effect only first order terms in beta matter. However, the
expressions given here are not restricted to beta~10^{-3}, but can be used at
much higher velocities. We demonstrate the robustness of these formulae,
illustrating the dependence of the kernel on beta, as well as the spherical
harmonic indices l and m.
