In this article we introduce the Riesz difference sequence space $r_p^q\left(\Delta^{B\alpha}\right)$ of fractional order $\alpha,$ defined by the composition of fractional backward difference operator $\Delta^{B\alpha}$ given by $(\Delta^{B\alpha}v)_k=\sum_{i=0}^{\infty}(-1)^i\frac{\Gamma(\alpha+1)}{i!\Gamma(\alpha-i+1)}v_{k-i}$ and the Riesz matrix $R^q.$ We give some topological properties, obtain the Schauder basis and determine the $\alpha$-, $\beta$- and $\gamma$- duals and investigate certain geometric properties of the space $r_p^q\left(\Delta^{B\alpha}\right)$. Finally, we characterize certain classes of compact operators on the space $r_p^q\left(\Delta^{B\alpha}\right)$ using Hausdorff measure of non-compactness.