  We calculate the Mellin moments of the $O(\alpha_s^2)$ coefficient functions
for the unpolarized and polarized fragmentation functions. They can be
expressed in terms of multiple finite harmonic sums of maximal weight {\sf w =
4}. Using algebraic and structural relations between harmonic sums one finds
that besides the single harmonic sums only three basic sums and their
derivatives w.r.t. the summation index contribute. The Mellin moments are
analytically continued to complex values of the Mellin variable. This
representation significantly reduces the large complexity being present in
$x$--space calculations and allow very compact and fast numerical
implementations.
