On the convergence difference sequences and the related operator norms

In this note, we discuss the definitions of the difference sequences defined earlier by Kızmaz (1981), Et and Çolak (1995), Malkowsky et al. (2007), Başar (2012), Baliarsingh (2013, 2015) and many others. Several authors have defined the difference sequence spaces and studied their various properties. It is quite natural to analyze the convergence of the corresponding sequences. As a part of this work, a convergence analysis of difference sequence of fractional order defined earlier is presented. It is demonstrated that the convergence of the fractional difference sequence is dynamic in nature and some of the results involved are also inconsistent. We provide certain stronger conditions on the primary sequence and the results due to earlier authors are substantially modified. Some illustrative examples are provided for each point of the modifications. Results on certain operator norms related to the difference operator of fractional order are also determined.