Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 104173, 6 pages
http://dx.doi.org/10.1155/2013/104173
Research Article
Fractional Sums and Differences with Binomial Coefficients
1Department of Mathematics, Faculty of Art and Sciencs, Çankaya University, Balgat, 06530 Ankara, Turkey
2Institute of Space Sciences, P.O. BOX MG-23, 76900 Magurele-Bucharest, Romania
3Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia
4Department of Mathematics, Texas A & M University, 700 University Boulevard, Kingsville, TX, USA
Received 31 March 2013; Accepted 3 May 2013
Academic Editor: Shurong Sun
Copyright © 2013 Thabet Abdeljawad 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
In fractional calculus, there are two approaches to obtain fractional derivatives. The first approach is by iterating the integral and then defining a fractional order by using Cauchy formula to obtain Riemann fractional integrals and derivatives. The second approach is by iterating the derivative and then defining a fractional order by making use of the binomial theorem to obtain Grünwald-Letnikov fractional derivatives. In this paper we formulate the delta and nabla discrete versions for left and right fractional integrals and derivatives representing the second approach. Then, we use the discrete version of the Q-operator and some discrete fractional dual identities to prove that the presented fractional differences and sums coincide with the discrete Riemann ones describing the first approach.