  We provide non-trivial checks of $\mathcal{N}=4, D=3$ mirror symmetry in a
large class of quiver gauge theories whose Type IIB (Hanany-Witten)
descriptions involve D3 branes ending on orbifold/orientifold 5-planes at the
boundary. From the M-theory perspective, such theories can be understood in
terms of coincident M2 branes sitting at the origin of a product of an A-type
and a D-type ALE (Asymtotically Locally Euclidean) space with G-fluxes.
Families of mirror dual pairs, which arise in this fashion, can be labeled as
$(A_{m-1},D_n)$, where $m$ and $n$ are integers. For a large subset of such
infinite families of dual theories, corresponding to generic values of $n\geq
4$, arbitrary ranks of the gauge groups and varying $m$, we test the
conjectured duality by proving the precise equality of the $S^3$ partition
functions for dual gauge theories in the IR as functions of masses and FI
parameters. The mirror map for a given pair of mirror dual theories can be read
off at the end of this computation and we explicitly present these for the
aforementioned examples. The computation uses non-trivial identities of
hyperbolic functions including certain generalizations of Cauchy determinant
identity and Schur's Pfaffian identity, which are discussed in the paper.
