Discrete Bessel Functions and Discrete Wave Equation

In this paper, we study four discrete Bessel functions which are solutions to the discretization of Bessel differential equations when the time derivative is replaced by the forward and the backward difference. We focus on discrete Bessel equations with the time derivative being the backward difference and derive their solutions: the discrete I-Bessel function \(\overline{I}_n^c(t)\) and the discrete J-Bessel function \(\overline{J}_n^c(t)\), \(t\in \mathbb {Z}\), \(n\in \mathbb {N}_0\). We then study transformation properties of those functions and describe their asymptotic behaviour as \(t\rightarrow \infty \) and as \(n\rightarrow \infty \). Moreover, we prove that the (unilateral) Laplace transform of \(\overline{I}_n^c\) and \(\overline{J}_n^c\) in the timescale \(T=\mathbb {Z}\) with the delta derivative being the backward difference equals the Laplace transform of classical I-Bessel and J-Bessel functions \(\mathcal {I}_n(cx)\) and \(\mathcal {J}_n(cx)\), respectively. As an application, we study the discrete wave equation on the integers in the timescale \(T=\mathbb {Z}\) and express its fundamental and general solution in terms of \(\overline{J}_n^c(t)\). Going further, we show that the first fundamental solution of this discrete wave equation oscillates with the exponentially decaying amplitude as time tends to infinity.