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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.

MSC:
46A45 Sequence spaces (including Köthe sequence spaces)
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References:
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