Discrete Bessel functions and discrete wave equation (Q6599759)

The authors propose discretizations of the classical Bessel differential equation where the time derivative is replaced by a forward or backward difference. Four discrete Bessel functions, solutions of the aforementioned discretized equations, are studied. For the discrete Bessel equation with the time derivative being the backward difference, the solutions \(\overline{J}^c_n(t)\) and \(\overline{I}^c_n(t)\) are derived (with \(n\in \mathbb{N}\), \(t\in \mathbb{Z}\), and \(c\in \mathbb{C}\setminus \{0\}\)). These functions are called by the authors discrete \(J\)-Bessel function and discrete \(I\)-Bessel function, respectively. Their asymptotic properties, as \(t\rightarrow\infty\) and \(n\rightarrow\infty\), are analyzed. Then, it is proved that the Laplace transform of \(\overline{J}^c_n(t)\) and \(\overline{I}^c_n(t)\) on an integer timescale equals the Laplace transform of the corresponding classical Bessel functions. The discrete wave equation (again on an integer timescale) is investigated, its fundamental solution is expressed in terms of \(\overline{J}^c_n(t)\), and some asymptotic properties for \(t\rightarrow\infty\) are established.