Banana integrals in configuration space

We reconsider the computation of banana integrals at different loops, by working in the configuration space, in any dimension. We show how the 2-loop banana integral can be computed directly from the configuration space representation, without the need to resort to differential equations, and we include the analytic extension of the diagram in the space of complex masses. We also determine explicitly the $\varepsilon$ expansion of the two loop banana integrals, for $d=j-2\varepsilon$, $j=2,3,4$. We also investigate the Picard-Fuchs equation systems for such integrals, and show that the same equations are nothing but a manifestation of certain standard recursive relations among Macdonald functions, and the associated Bessel-type second-order differential equation. Therein, we show how in the same way, from such recursive relations, one generalizes easily the differential equations for banana integrals with an arbitrary number of loops, by means of elementary steps. We finally determine a (non manifestly symmetric) expression for the three loop banana integral.