  We report on the status an ab initio computation of the time-like splitting
functions at next-to-next-to-leading order in QCD. Time-like splitting
functions govern the collinear kinematics of inclusive hadron production in
$e^+e^-$ annihilation and the evolution of the parton fragmentation
distributions. Current knowledge about them at three loops has been inferred by
means of crossing symmetry from their related space-like counterparts, the
deep-inelastic structure functions and parton densities. In this approach
certain parts of the off-diagonal quark-gluon splitting function are left
undetermined, which calls for an independent calculation from first principles.
We outline the method for calculating master integrals from differential
equations which are required to attack the problem.
