  We recover in QCD an amazingly simple relationship between the anomalous
dimensions, resummed through next-to-next-to-leading-logarithmic order, in the
Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equations for the first
Mellin moments $D_{q,g}(\mu^2)$ of the quark and gluon fragmentation functions,
which correspond to the average hadron multiplicities in jets initiated by
quarks and gluons, respectively. This relationship, which is independent of the
number of quark flavors, dramatically improves previous treatments by allowing
for an exact solution of the evolution equations. So far, such relationships
have only been known from supersymmetric QCD, where $C_F/C_A=1$. This also
allows us to extend our knowledge of the ratio $D_g^-(\mu^2)/D_q^-(\mu^2)$ of
the minus components by one order in $\sqrt{\alpha_s}$. Exploiting available
next-to-next-to-next-to-leading-order information on the ratio
$D_g^+(\mu^2)/D_q^+(\mu^2)$ of the dominant plus components, we fit the world
data of $D_{q,g}(\mu^2)$ for charged hadrons measured in $e^+e^-$ annihilation
to obtain $\alpha_s^{(5)}(M_Z)=0.1205\genfrac{}{}{0pt}{}{+0.016}{-0.0020}$.
