  We have derived the coefficients of the highest three 1/x-enhanced small-x
logarithms of all timelike splitting functions and the coefficient functions
for the transverse fragmentation function in one-particle inclusive e^+e^-
annihilation at (in principle) all orders in massless perturbative QCD. For the
longitudinal fragmentation function we present the respective two highest
contributions. These results have been obtained from KLN-related decompositions
of the unfactorized fragmentation functions in dimensional regularization and
their structure imposed by the mass-factorization theorem. The resummation is
found to completely remove the huge small-x spikes present in the fixed-order
results for all quantities above, allowing for stable results down to very
small values of the momentum fraction and scaling variable x. Our calculations
can be extended to (at least) the corresponding as^n ln^(2n-l) x contributions
to the above quantities and their counterparts in deep-inelastic scattering.

High Energy Physics - Phenomenology