  We analyse power corrections to longitudinal and transverse fragmentation
processes in e^+ e^- annihilation, based on the assumption of ultraviolet
dominance of power corrections. Under this assumption, we determine the
dependence of power corrections on the scaling variable x from the infrared
renormalon asymptotics of leading power coefficient functions. Our results
suggest that the longitudinal and transverse gluon fragmentation coefficient
functions receive corrections of order 1/(x Q)^2. The power expansion breaks
down at x<\Lambda/Q and has to be resummed. This resummation leads to 1/Q
corrections to the longitudinal and transverse cross section, which cancel for
the total cross section. We provide a simple parametrization of the x
dependence of 1/Q^2 corrections to fragmentation processes and investigate
perturbative corrections to the longitudinal cross section in higher orders, in
view of a determination of the strong coupling.
