Some Euler spaces of difference sequences of order m*

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 er0, erc, and er∞, respectively. The main purpose of this article is to introduce the spaces er0(Δm), erc(Δm), and er∞(Δm) consisting of all sequences whose mth order differences are in the Euler spaces er0, erc, and er∞, respectively. Moreover, the authors give some topological properties and inclusion relations, and determine the α-, β-, and γ-duals of the spaces er0(Δm), erc(Δm), and er∞(Δm), and the Schauder basis of the spaces er0(Δm), erc(Δm). The last section of the article is devoted to the characterization of some matrix mappings on the sequence space erc(Δm).