  We evaluate, analytically, a specific class of eighth--order and tenth--order
QED contributions to the anomalous magnetic moment of the muon. They are
generated by Feynman diagrams involving lowest order vacuum polarization
insertions of leptons $l=e,\mu$, and $\tau$. The results are given in the form
of analytic expansions in terms of the mass ratios $m_e/m_\mu$ and
$m_\mu/m_\tau$. We compute as many terms as required by the error induced by
the present experimental uncertainty on the lepton masses. We show how the
Mellin--Barnes integral representation of Feynman parametric integrals allows
for an easy analytic evaluation of as many terms as wanted in these expansions
and how its underlying algebraic structure generalizes the standard
renormalization group properties. We also discuss the generalization of this
technique to the case where two independent mass ratios appear. Comparison with
previous numerical and analytic evaluations made in the literature, whenever
pertinent, are also made.
