  Long-term memory is a feature observed in systems ranging from neural
networks to epidemiological models. The memory in such systems is usually
modeled by the time delay. Furthermore, the nonlocal operators, such as the
"fractional order difference" can also have a long-time memory. Therefore, the
fractional difference equations with delay are an appropriate model in a range
of systems. Even so, there are not many detailed studies available related to
the stability analysis of fractional order systems with delay. In this work, we
derive the stability conditions for linear fractional difference equations with
a delay term $\tau$. We have given detailed stability analysis for the cases
$\tau=1$ and $\tau=2$. The results are extended to nonlinear maps.
