# zbMATH — the first resource for mathematics

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)
Full Text:
##### 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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.