We provide non-trivial checks of 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
(Am−1​,Dn​), where m and n are integers. For a large subset of such
infinite families of dual theories, corresponding to generic values of n≥4, arbitrary ranks of the gauge groups and varying m, we test the
conjectured duality by proving the precise equality of the S3 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.Comment: 45 pages, 9 figure