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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:
 [1] COOKE R. G.: Infinite Matrices and Sequence Spaces. MacMillan, London, 1950. · Zbl 0040.02501 [2] ET M.-COLAK R.: On some generalized difference spaces. Soochow J. Math. 21 (1995), 377-386. · Zbl 0841.46006 [3] KAMTHAN P. K.-GUPTA M.: Sequence Spaces and Series. Marcel Dekker Inc., New York, 1981. · Zbl 0447.46002 [4] KIZMAZ H.: On certain sequence spaces. Canad. Math. Bull. 24 (1981), 169-176. · Zbl 0454.46010 [5] TRIPATHY B. C.: A class of difference sequence sequences related to the $$p$$-normed space $$\ell^p$$. Demonstratio Math. 36 (2003), 867-872. · Zbl 1042.40001 [6] TRIPATHY B. C.: On some class of difference paranormed sequence spaces associated with multiplier sequences. Int. J. Math. Math Sci. 2 (2003), 159-166. · Zbl 1064.40002 [7] TRIPATHY B. C.-ESI A.: A new type of difference sequence spaces. Internat. J. Sci. Tech. 1 (2006), 11-14.
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