This paper concerns with the asymptotic behavior of exponential functions on time scales. A time scale \({\mathbb T}\) is a nonempty closed subset of \({\mathbb R}\) and the calculus on time scales enables to work simultaneously with the classical continuous-time (= differential, \({\mathbb T}={\mathbb R}\), \({\mathbb T}=[a,b]\), …) calculus and the discrete-time (= difference, \({\mathbb T}={\mathbb Z}\), \({\mathbb T}=\{0,1,\dots,N\}\), …) calculus, as well as with any other time scale as mentioned above. An exponential function is a nontrivial solution of the first order dynamic equation \(y^\Delta=\lambda\,y\), where \(y:{\mathbb T}\to{\mathbb R}\), \(y^\Delta\) is the time scale (or \(\Delta\)-) derivative, and \(\lambda\) is a constant satisfying a certain regressivity condition.
The authors calculate these exponential functions for many different time scales, such as \({\mathbb R}\), \({\mathbb N}_0\), \({\mathbb N}_0^p\), \(q^{{\mathbb N}_0}\), etc., and determine their behavior – in principal terms – as \(t\) becomes arbitrarily large (hence only time scales which are unbounded above are considered). As computational techniques, the authors use explicit computations, product representations, dynamic equations (or functional equations) approach, and certain asymptotic exponential classes. In the final section the authors present the time scale version of the Harris algorithm (a generalized Putzer algorithm) for computing the time scale matrix exponential functions, i.e., the solutions of the matrix dynamic equation \(y^\Delta=A\,y\) with constant matrix coefficient \(A\).
This paper is an important and elegant contribution to the theory of dynamic equations on time scales. Moreover, anyone interested in the asymptotic theory of differential/difference equations will find this paper useful.