  We present the third-order contributions to the quark-gluon and gluon-quark
timelike splitting functions for the evolution of fragmentation functions in
perturbative QCD. These quantities have been derived by studying physical
evolution kernels for photon- and Higgs-exchange structure functions in deep
inelastic scattering and their counterparts in semi-inclusive annihilation,
together with constraints from the momentum sum rule and the supersymmetric
limit. For this purpose we have also calculated the second-order coefficient
functions for one-hadron inclusive Higgs decay in the heavy-top limit. A
numerically tolerable uncertainty remains for the quark-gluon splitting
function, which does not affect the endpoint logarithms for small and large
momentum fractions. We briefly discuss these limits and illustrate the
numerical impact of the third-order corrections. Compact and accurate
parametrizations are provided for all third-order timelike splitting functions.
