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 $\lambda \in {c_{0},c,\ell_{\infty},\ell_1}$ .