  In this paper, we study the $\ell^p$-maximal regularity for the fractional
difference equation with finite delay: \begin{equation*} \ \ \ \ \ \ \ \
\left\{\begin{array}{cc}
  \Delta^{\alpha}u(n)=Au(n)+\gamma u(n-\lambda)+f(n), \ n\in \mathbb N_0,
\lambda \in \mathbb N, \gamma \in \mathbb R;
  u(i)=0,\ \ i=-\lambda, -\lambda+1,\cdots, 1, 2,
  \end{array} \right. \end{equation*} where $A$ is a bounded linear operator
defined on a Banach space $X$, $f:\mathbb N_0\rightarrow X$ is an $X$-valued
sequence and $2<\alpha<3$. We introduce an operator theoretical method based on
the notion of $\alpha$-resolvent sequence of bounded linear operators, which
gives an explicit representation of solution. Further, using Blunck's
operator-valued Fourier multipliers theorems on $\ell^p(\mathbb{Z}; X)$, we
completely characterize the $\ell^p$-maximal regularity of solution when $1 < p
< \infty$ and $X$ is a UMD space.
