  We investigate numerically the problem of few (one, two) noninteracting
spin$-1/2$ fermions in a shallow harmonic trap coupled via contact repulsive
interactions to a uniform one-dimensional bath of lattice bosons, described by
the Bose-Hubbard model. Through extensive density-matrix renormalization group
calculations, we extract the binding energy and the effective mass of
quasiparticles, including dressed impurities (polarons) and their two-body
bound states (bipolarons), emerging from the effective non-local Casimir
interaction between the impurities. We show that the mixture exhibits rather
different pairing behaviors depending on the singlet \textit{vs}.~triplet spin
state configurations of the two fermions. For opposite spin states, bipolarons
are found for any finite value of the impurity-bath coupling. In particular, in
the strong coupling regime their binding energy reduces to that of a single
polaron, provided the boson-boson repulsion is not too weak. For equal spin
states, we show that bipolarons emerge only beyond a critical strength of the
Bose-Fermi interaction and their effective mass grows rapidly approaching the
strong coupling regime.
