Some Euler spaces of difference sequences of order m

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Kizmaz [13] studied the difference sequence spaces ℓ ∞(Δ), c(Δ), and c0(Δ). Several article dealt with the sets of sequences of m-th order difference of which are bounded, convergent, or convergent to zero. Altay and Başar [5] and Altay, Başar, and Mursaleen [7] introduced the Euler sequence spaces e r 0, e r c , and e r ∞, respectively. The main purpose of this article is to introduce the spaces e r 0(Δ m ), e r c (Δ m ), and e r ∞(Δ m ) consisting of all sequences whose m th order differences are in the Euler spaces e r 0, e r c , and e r ∞, respectively. Moreover, the authors give some topological properties and inclusion relations, and determine the α-, β-, and γ-duals of the spaces e r 0(Δ m ), e r c (Δ m ), and e r ∞(Δ m ), and the Schauder basis of the spaces e r 0(Δ m ), e r c (Δ m ). The last section of the article is devoted to the characterization of some matrix mappings on the sequence space e r c (Δ m ).