  We determine the two-loop 'time-like' Altarelli-Parisi splitting functions,
appearing in the next-to-leading order Q^2-evolution equations for
fragmentation functions, via analytic continuation of the corresponding
'space-like' splitting functions for the evolution of parton distributions. We
do this for the case of unpolarized fragmentation functions and - for the first
time - also for the functions describing the fragmentation of a longitudinally
polarized parton into a longitudinally polarized spin-1/2 hadron such as a
Lambda baryon. Our calculation is based on the method proposed and employed by
Curci, Furmanski and Petronzio in the unpolarized case in which we confirm
their results.
