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On some new type generalized difference sequence spaces. (English) Zbl 1164.46004
Let \(\ell _{\infty },c\) and \(c_{0}\) denote the spaces of all bounded, convergent and null sequences \(x=(x_{k})\) with complex terms, respectively, with the norm \(\left \| x\right \| =\sup _{k }\left | {x_{k}}\right | \). The idea of difference sequence spaces, \(X(\Delta)=\{x=(x_{k})\:\Delta x\in X\}\), where \(X=\ell _{\infty }, c\) and \(c_{0}\) was introduced by H. Kizmaz [Can. Math. Bull. 24, 169–176 (1981; Zbl 0454.46010)], and since then this subject has been studied and generalized by various mathematicians. In this study, the authors introduce some new type of generalized difference sequence spaces \[ X\bigl (\Delta _{m}^{n}\bigr)=\bigl \{x=(x_{k}):\Delta _{m}^{n}x=\bigl (\Delta _{m}^{n}x_{k}\bigr)\in Z\bigr \}, \] for fixed \(m,n\in \mathbb N\) and \(Z=\ell _{\infty }, \,c\) and \(c_{0}\), where \(\Delta _{m}^{n}x=\bigl (\Delta _{m}^{n}x_{k}\bigr)=\bigl (\Delta _{m}^{n-1}x_{k}-\Delta _{m}^{n-1}x_{k+m}\bigr)\) and \(\Delta _{m}^{0}x_{k}=x_{k}\) for all \(k\in \mathbb N\). The generalized difference notion has the following binomial representation: \[ \Delta _{m}^{n}x_{k}=\sum \limits _{v=0}^{n}(-1)^{v}(_{v}^{n})x_{k+mv}\qquad \text{for \,all}\quad k\in \mathbb N. \]
Some properties of the defined spaces are described. For example, it is proved that all spaces defined here are Banach spaces with the norm \(\left \| x\right \| _{\Delta _{m}^{n}}=\sum \limits _{i=1}^{r}\left | {x_{i}}\right | +\sup _k\left | \Delta _{m}^{n}x_{k}\right | \), where \(r=mn\) for \(m\geq 1, \,n\geq 1; \,r=n\) for \(m=1\) and \(r=m\) for \(n=1\) and are not symmetric, monotone and solid spaces in general. In addition, the inclusion relations involving these spaces are obtained. The results generalize some previously known results.

46A45 Sequence spaces (including Köthe sequence spaces)
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