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On the classes of fractional order difference sequence spaces and their matrix transformations. (English) Zbl 1328.46002
Summary: The main purpose of the present article is to introduce the classes of generalized fractional order difference sequence spaces \(\ell_\infty(\operatorname{\Gamma}, \operatorname{\Delta}^{\widetilde{\alpha}}, p), c_0(\operatorname{\Gamma}, \operatorname{\Delta}^{\widetilde{\alpha}}, p)\) and \(c(\operatorname{\Gamma}, \operatorname{\Delta}^{\widetilde{\alpha}}, p)\) by defining the fractional difference operator \(\operatorname{\Delta}^{\widetilde{\alpha}} x_k = \sum_{i = 0}^\infty(- 1)^i \frac{\operatorname{\Gamma}(\widetilde{\alpha} + 1)}{i! \operatorname{\Gamma}(\widetilde{\alpha} - i + 1)} x_{k + i}\), where \(\widetilde{\alpha}\) is a positive proper fraction and \(k \in \mathbb{N} = \{1, 2, 3 \ldots . \}\). Results concerning the linearity and various topological properties of these spaces are established and also the alpha-, beta-, gamma- and \(N\)-duals of these spaces are obtained. The matrix transformations from these classes into Maddox spaces are also characterized. Throughout the article we use the notation \(\operatorname{\Gamma}(n)\) as the Gamma function of \( n\), defined by an improper integral \(\operatorname{\Gamma}(n) = \int_0^\infty e^{- t} t^{n - 1} {dt}\), where \(n \notin \{0, - 1, - 2, \ldots \}\) and \(\operatorname{\Gamma}(n + 1) = n \operatorname{\Gamma}(n)\).

MSC:
46A45 Sequence spaces (including Köthe sequence spaces)
40A05 Convergence and divergence of series and sequences
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