  It is well-known that the fundamental solution of $$ u_t(n,t)=
u(n+1,t)-2u(n,t)+u(n-1,t), \quad n\in\mathbb{Z}, $$ with $u(n,0) =\delta_{nm}$
for every fixed $m \in\mathbb{Z}$, is given by $u(n,t) = e^{-2t}I_{n-m}(2t)$,
where $I_k(t)$ is the Bessel function of imaginary argument. In other words,
the heat semigroup of the discrete Laplacian is described by the formal series
$$ W_tf(n) = \sum_{m\in\mathbb{Z}} e^{-2t} I_{n-m}(2t) f(m). $$ By using
semigroup theory, this formula allows us to analyze some operators associated
with the discrete Laplacian. In particular, we obtain the maximum principle for
the discrete fractional Laplacian, weighted $\ell^p(\mathbb{Z})$-boundedness of
conjugate harmonic functions, Riesz transforms and square functions of
Littlewood-Paley.
  Interestingly, it is shown that the Riesz transforms coincide essentially
with the so called discrete Hilbert transform defined by D. Hilbert at the
beginning of the XX century. We also see that these Riesz transforms are limits
of the conjugate harmonic functions.
  The results rely on a careful use of several properties of Bessel functions.
