Euler-Riesz Difference Sequence Spaces

Abstract

Ba\c{s}ar and Braha \cite{braha-basar-2016}, introduced the

sequence spaces $\ell_\infty$, $c$ and $c_0$  of Euler- Ces\'{a}ro

bounded, convergent and null difference sequences and studied

their some properties. The main purpose of this study is to

introduce 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

$\Delta$. Furthermore, the inclusions

$\ell_\infty\subset{[\ell_\infty]}_{e.r}, c\subset {[c]}_{e.r}$

and $c_0\subset{[c_0]}_{e.r}$ strictly hold, the basis of the

sequence spaces ${[c_0 ]}_(e.r)$ and ${[c]}_(e.r)$ is constucted

and alpha-, beta- and gamma-duals of these spaces are determined.

Finally, the classes of matrix transformations from the Euler-

Riesz difference sequence spaces to the spaces $\ell_\infty, c$

and $c_0$ are characterized. 

Keywords

• Euler sequence spaces,Riesz sequence spaces,matrix transformations

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