On some new difference sequence spaces of fractional order

Abstract

Let ${\Delta }^{(\alpha )}$ denote the fractional difference operator. In this paper, we define new difference sequence spaces $c_0(\Gamma ,{\Delta }^{(\alpha )},u)$ and $c(\Gamma ,{\Delta }^{(\alpha )},u)$. Also, the $\beta -$ dual of the spaces $c_0(\Gamma ,{\Delta }^{(\alpha )},u)$ and $c(\Gamma ,{\Delta }^{(\alpha )},u)$ are determined and calculated their Schauder basis. Furthermore, the classes $(\mu (\Gamma ,{\Delta }^{(\alpha )},u):\lambda )$ where $\mu \in \{c_0,c\}$ and $\lambda \in \{c_0,c,{\ell }_{\infty },{\ell }_1\}$ .

14 pages