  We define and study some properties of the fractional powers of the discrete
Laplacian $$(-\Delta_h)^s,\quad\hbox{on}~\mathbb{Z}_h = h\mathbb{Z},$$ for
$h>0$ and $0<s<1$. A comparison between our fractional discrete Laplacian and
the \textit{discretized} continuous fractional Laplacian as $h\to0$ is carried
out. We get estimates in $\ell^\infty$ for the error of the approximation in
terms of $h$ under minimal regularity assumptions. Moreover, we provide a
pointwise formula with an explicit kernel and deduce H\"older estimates for
$(-\Delta_h)^s$. A study of the negative powers (or discrete fractional
integral) $(-\Delta_h)^{-s}$ is also sketched. Our analysis is mainly performed
in dimension one. Nevertheless, we show certain asymptotic estimates for the
kernel in dimension two that can be extended to higher dimensions. Some
examples are plotted to illustrate the comparison in both one and two
dimensions.
