  In this paper, we study discrete Bessel functions which are solutions to the
discretization of Bessel differential equations when the forward and the
backward difference replace the time derivative. We focus on the discrete
Bessel equations with the backward difference and derive their solutions. We
then study the transformation properties of those functions, describe their
asymptotic behaviour and compute Laplace transform. As an application, we study
the discrete wave equation on the integers in timescale $T=\mathbb{Z}$ and
express its fundamental and general solution in terms of the discrete
$J$-Bessel function. Going further, we show that the first fundamental solution
of this equation oscillates with the exponentially decaying amplitude as time
tends to infinity.
