  Four dimensional $\mathcal{N}=2$ Argyres-Douglas theories have been recently
conjectured to be described by $\mathcal{N}=1$ Lagrangian theories. Such
models, once reduced to 3d, should be mirror dual to Lagrangian $\mathcal{N}=4$
theories. This has been numerically checked through the matching of the
partition functions on the three sphere. In this article, we provide an
analytic derivation for this result in the $A_n$ case via hyperbolic
hypergeometric integrals. We study the $D_4$ case as well, commenting on some
open questions and possible resolutions. In the second part of the paper we
discuss other integral identities leading to the matching of the partition
functions in 3d dual pairs involving higher monopole superpotentials.
