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\({[\ell_p]}_{e.r}\) Euler-Riesz difference sequence spaces. (English) Zbl 1499.46012

Summary: F. Başar and N. L. Braha [Tamkang J. Math. 47, No. 4, 405–420 (2016; Zbl 1373.46003)], introduced the sequence spaces \(\breve{\ell}_\infty, \breve{c}\) and \(\breve{c}_0\) of Euler-Cesàro bounded, convergent and null difference sequences and studied their some properties. Then, in [“Euler-Riesz difference sequence spaces”, Turk. J. Math. Comput. Sci. 7, 63–72 (2017)], we introduced the sequence spaces \({[\ell_\infty]}_{e.r}, {[c]}_{e.r}\) and \({[c_0]}_{e.r}\) of Euler-Riesz bounded, convergent and null difference sequences by using the composition of the Euler mean \(E_1\) and Riesz mean \(R_q\) with backward difference operator \(\Delta \). The main purpose of this study is to introduce the sequence space \({[\ell_p]}_{e.r}\) of Euler-Riesz \(p\)-absolutely convergent series, where \(1 \leq p <\infty \), difference sequences by using the composition of the Euler mean \(E_1\) and Riesz mean \(R_q\) with backward difference operator \(\Delta \). Furthermore, the inclusion \(\ell_p\subset{[\ell_p]}_{e.r}\) hold, the basis of the sequence space \({[\ell_p]}_{e.r}\) is constructed and \(\alpha\)-, \(\beta\)- and \(\gamma\)-duals of the space are determined. Finally, the classes of matrix transformations from the \({[\ell_p]}_{e.r}\) Euler-Riesz difference sequence space to the spaces \(\ell_\infty, c\) and \(c_0\) are characterized. We devote the final section of the paper to examine some geometric properties of the space \({[\ell_p]}_{e.r}\).

MSC:

46A45 Sequence spaces (including Köthe sequence spaces)
40C05 Matrix methods for summability
40A05 Convergence and divergence of series and sequences

Citations:

Zbl 1373.46003
Full Text: DOI

References:

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