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Some new difference sequence spaces of fractional order and their dual spaces. (English) Zbl 1300.46004

Summary: The main objective of the present paper is to introduce certain difference sequence spaces of fractional order and investigate their topological structures as well as some interesting results concerning the fractional difference operator \(\Delta^\alpha\). In this paper, we define some new difference sequence spaces such as \(\ell_\infty(\Gamma, \Delta^\alpha, u)\), \(c_0(\Gamma, \Delta^\alpha, u)\) and \(c(\Gamma, \Delta^\alpha, u)\) by introducing a fractional difference operator \(\Delta^\alpha\) and, for a positive fraction \(\alpha\), the difference operator \(\Delta^\alpha\) is defined by \(\Delta^\alpha(\chi_k) = \sum_{i=0}^\infty(-1)^i \frac{\Gamma(\alpha + 1)}{i!\Gamma(\alpha - i + 1)}\chi_{k+i}\). Also, we establish their \(\alpha\)-, \(\beta\)- and \(\gamma\)-duals.

MSC:

46A45 Sequence spaces (including Köthe sequence spaces)
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