TY - JOUR
T1 - ${[\ell_p]}_{e.r}$ Euler-Riesz Difference Sequence Spaces
AU - Ellidokuzoğlu , Hacer Bilgin
AU - Demiriz , Serkan
JO - Analysis in Theory and Applications
VL - 4
SP - 557
EP - 571
PY - 2021
DA - 2021/11
SN - 37
DO - http://doi.org/10.4208/ata.OA-2017-0068
UR - https://global-sci.org/intro/article_detail/ata/19964.html
KW - Composition of summability methods, Riesz mean of order one, Euler mean of order one, backward difference operator, sequence space, BK space, Schauder basis, $\beta-$duals, matrix transformations.
AB - <p style="text-align: justify;">Başar and Braha [1], introduced the sequence spaces $\breve{\ell}_\infty$, $\breve{c}$ and $\breve{c}_0$ of Euler-Cesáro bounded, convergent and null difference sequences and studied their some properties. Then, in [2], 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 $1 \leq p &lt;\infty$, 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&nbsp; $\alpha-$, $\beta-$ 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 ${[\ell_p]}_{e.r}$.</p>