Euler-Riesz Difference Sequence Spaces

Abstract

Ba\c{s}ar and Braha \cite{braha-basar-2016}, introduced thesequence spaces $\ell_\infty$, $c$ and $c_0$ of Euler- Ces\'{a}robounded, convergent and null difference sequences and studiedtheir some properties. The main purpose of this study is tointroduce the sequence spaces ${[\ell_\infty]}_{e.r},{[c]}_{e.r}$and ${[c_0]}_{e.r}$ of Euler- Riesz bounded, convergent and nulldifference 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 thesequence spaces ${[c_0 ]}_(e.r)$ and ${[c]}_(e.r)$ is constuctedand 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.