Paranormed Riesz Difference Sequence Spaces of Fractional Order

Taja Yaying

In this article we introduce paranormed Riesz difference sequence spaces of fractional order $\alpha,$ $r_0^t\left(p, \Delta^{(\alpha)}\right),$ $r_c^t\left(p, \Delta^{(\alpha)}\right)$ and $r_{\infty}^t\left(p, \Delta^{(\alpha)}\right) $ defined by the composition of fractional difference operator $\Delta^{(\alpha)},$ defined by $(\Delta^{(\alpha)}x)_k=\sum\limits_{i=0}^{\infty}(-1)^i\frac{\Gamma(\alpha+1)}{i!\Gamma(\alpha-i+1)}x_{k-i},$ and Riesz mean matrix $R^t.$ We give some topological properties, obtain the Schauder basis and determine the $\alpha$-, $\beta$- and $\gamma$- duals of the new spaces. Finally, we characterize certain matrix classes related to these new spaces.