Some q-fractional order difference sequence spaces

This paper intends to develop a $q$-difference operator $\nabla^{(\gamma)}_q$ of fractional order $\gamma$, and give several intriguing properties of this new difference operator. Our main focus remains on the construction of sequence spaces $\ell_p(\nabla^{(\gamma)})$ and $\ell_\infty (\nabla^{(\gamma)})$, at the same time comparing these spaces with those already exist in the literature. Apart from obtaining Schauder basis, we determine $\alpha$-, $\beta$-, and $\gamma$-duals of the newly defined spaces. A section is also devoted for characterizing matrix classes $(\ell_p(\nabla^{(\gamma)}),\mathfrak X),$ where $\mathfrak X$ is any of the spaces $\ell_\infty,$ $c,$ $c_0$ and $\ell_1$.