  We study the splitting functions for the evolution of fragmentation
distributions and the coefficient functions for single-hadron production in
semi-inclusive electron-positron annihilation in massless perturbative QCD for
small values of the momentum fraction and scaling variable x, where their
fixed-order approximations are completely destabilized by huge double
logarithms of the form alpha_s^n 1/x ln^(2n-a) x. Complete analytic all-order
expressions in Mellin-N space are presented for the resummation of these terms
at the next-to-next-to-leading logarithmic accuracy. The poles for the first
moments, related to the evolution of hadron multiplicities, and the small-x
instabilities of the next-to-leading order splitting and coefficient functions
are removed by this resummation, which leads to an oscillatory small-x
behaviour and functions that can be used at N=1 and down to extremely small
values of x. First steps are presented towards extending these results to the
higher accuracy required for an all-x combination with the state-of-the-art
next-to-next-to-leading order large-x results.
