On a fractional difference operator

In the present article, a set of new difference sequence spaces of fractional order has been introduced and subsequently, an application of these spaces, the notion of the derivatives and the integrals of a function to the case of non-integer order have been generalized. Certain results involving the unusual and non-uniform behavior of the corresponding difference operator have been investigated and also been verified by using some counter examples. We also verify these unusual and non-uniform behaviors by studying the geometry of fractional calculus. (C) 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).