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We study the splitting functions for the evolution of fragmentationdistributions and the coefficient functions for single-hadron production insemi-inclusive electron-positron annihilation in massless perturbative QCD forsmall values of the momentum fraction and scaling variable x, where theirfixed-order approximations are completely destabilized by huge doublelogarithms of the form alpha_s^n 1/x ln^(2n-a) x. Complete analytic all-orderexpressions in Mellin-N space are presented for the resummation of these termsat the next-to-next-to-leading logarithmic accuracy. The poles for the firstmoments, related to the evolution of hadron multiplicities, and the small-xinstabilities of the next-to-leading order splitting and coefficient functionsare removed by this resummation, which leads to an oscillatory small-xbehaviour and functions that can be used at N=1 and down to extremely smallvalues of x. First steps are presented towards extending these results to thehigher accuracy required for an all-x combination with the state-of-the-artnext-to-next-to-leading order large-x results.

Resummed small-x and first-moment evolution of fragmentation functions in perturbative QCD