Abstract
Abstract The Laplace transform is important for exact solutions of linear differential equations and frequency response analysis methods. In comparison with the continuous–time systems, less results can be available for fractional difference equations. This study provides some fundamental results of two kinds of fractional difference equations by use of the Laplace transform. Some discrete Mittag–Leffler functions are defined and their Laplace transforms are given. Furthermore, a class of variable–order and short memory linear fractional difference equations are proposed and the exact solutions are obtained.
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