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Discrete Dynamics in Nature and Society
Volume 2013, Article ID 406910, 12 pages
Research Article

On Delta and Nabla Caputo Fractional Differences and Dual Identities

1Department of Mathematics and Physical Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia
2Department of Mathematics, Çankaya University, 06530 Ankara, Turkey

Received 13 January 2013; Accepted 14 June 2013

Academic Editor: Shurong Sun

Copyright © 2013 Thabet Abdeljawad. 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.


We investigate two types of dual identities for Caputo fractional differences. The first type relates nabla and delta type fractional sums and differences. The second type represented by the Q-operator relates left and right fractional sums and differences. Two types of Caputo fractional differences are introduced; one of them (dual one) is defined so that it obeys the investigated dual identities. The relation between Riemann and Caputo fractional differences is investigated, and the delta and nabla discrete Mittag-Leffler functions are confirmed by solving Caputo type linear fractional difference equations. A nabla integration by parts formula is obtained for Caputo fractional differences as well.