  This work focuses on the development of a new class of high-order accurate
methods for multirate time integration of systems of ordinary differential
equations. Unlike other recent work in this area, the proposed methods support
mixed implicit-explicit (IMEX) treatment of the slow time scale. In addition to
allowing this slow time scale flexibility, the proposed methods utilize a
so-called `infinitesimal' formulation for the fast time scale through
definition of a sequence of modified `fast' initial-value problems, that may be
solved using any viable algorithm. We name the proposed class as
implicit-explicit multirate infinitesimal generalized-structure additive
Runge--Kutta (IMEX-MRI-GARK) methods. In addition to defining these methods, we
prove that they may be viewed as specific instances of GARK methods, and derive
a set of order conditions on the IMEX-MRI-GARK coefficients to guarantee both
third and fourth order accuracy for the overall multirate method. Additionally,
we provide three specific IMEX-MRI-GARK methods, two of order three and one of
order four. We conclude with numerical simulations on two multirate test
problems, demonstrating the methods' predicted convergence rates and comparing
their efficiency against both legacy IMEX multirate schemes and recent third
and fourth order implicit MRI-GARK methods.
