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The three-loop QED mass-dependent contributions to the $g-2$ of each of thecharged leptons with two internal closed fermion loops, sometimes called$A^{(6)}_3\left(\frac{m_1}{m_2}, \frac{m_1}{m_3}\right)$ in the $g-2$literature, is revisited using the Mellin-Barnes (MB) representation technique.Results for the muon and $\tau$ lepton anomalous magnetic moments$A^{(6)}_{3,\mu}$ and $A^{(6)}_{3,\tau}$, which were known as series expansionsin the lepton mass ratios up to the first few terms only, are extended to theirexact expressions. The contribution to the anomalous magnetic moment of theelectron $A^{(6)}_{3,e}$ is also explicitly given in closed form. In additionto this, we show that the different series representations derived from the MBrepresentation collectively converge for all possible values of the masses.Such unexpected behavior is related to the fact that these series bring intoplay double hypergeometric series that belong to a class of Kamp\'e de F\'erietseries which we prove to have the same simple convergence and analyticcontinuation properties as the Appell $F_1$ double hypergeometric series.

The three-loop QED contributions to the $g-2$ of charged leptons with two internal fermion loops and a class of Kamp\'e de F\'eriet series