  We give a detailed account of the phenomenology of all-order resummations of
logarithmically enhanced contributions at small momentum fraction of the
observed hadron in semi-inclusive electron-positron annihilation and the
time-like scale evolution of parton-to-hadron fragmentation functions. The
formalism to perform resummations in Mellin moment space is briefly reviewed,
and all relevant expressions up to next-to-next-to-leading logarithmic order
are derived, including their explicit dependence on the factorization and
renormalization scales. We discuss the details pertinent to a proper numerical
implementation of the resummed results comprising an iterative solution to the
time-like evolution equations, the matching to known fixed-order expressions,
and the choice of the contour in the Mellin inverse transformation. First
extractions of parton-to-pion fragmentation functions from semi-inclusive
annihilation data are performed at different logarithmic orders of the
resummations in order to estimate their phenomenological relevance. To this
end, we compare our results to corresponding fits up to fixed,
next-to-next-to-leading order accuracy and study the residual dependence on the
factorization scale in each case.
