Abstract
We show the new relationship [1] between the anomalous dimensions, resummed through next-to-next-to-leading-logarithmic order (NNNLO), in the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) 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 strongly improves previous treatments by allowing for an exact solution of the evolution equations. The fit of the world data of $${{D}_{{q,g}}}({{\mu }^{2}})$$ for charged hadrons measured in $${{e}^{ + }}{{e}^{ - }}$$ annihilation leads to $$\alpha _{s}^{{(5)}}({{M}_{Z}}) = 0.1205\begin{array}{*{20}{c}} { + 0.0016} \\ {0.0020} \end{array}.$$
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