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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 565067, 21 pages
http://dx.doi.org/10.1155/2011/565067
Research Article

Discrete Mittag-Leffler Functions in Linear Fractional Difference Equations

Institute of Mathematics, Faculty of Mechanical Engineering, Technická 2, 616 69 Brno, Czech Republic

Received 7 January 2011; Accepted 23 April 2011

Academic Editor: Yuri V. Rogovchenko

Copyright © 2011 Jan Čermák et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper investigates some initial value problems in discrete fractional calculus. We introduce a linear difference equation of fractional order along with suitable initial conditions of fractional type and prove the existence and uniqueness of the solution. Then the structure of the solutions space is discussed, and, in a particular case, an explicit form of the general solution involving discrete analogues of Mittag-Leffler functions is presented. All our observations are performed on a special time scale which unifies and generalizes ordinary difference calculus and 𝑞-difference calculus. Some of our results are new also in these particular discrete settings.