Some q-fractional order difference sequence spaces

This paper intends to develop a $q$-difference operator ${\mathrm{\nabla }}_q^{(\gamma )}$ of fractional order $\gamma$, and give several intriguing properties of this new difference operator. Our main focus remains on the construction of sequence spaces ${\mathrm{\ell }}_p({\mathrm{\nabla }}^{(\gamma )})$ and ${\mathrm{\ell }}_{\mathrm{\infty }}({\mathrm{\nabla }}^{(\gamma )})$, 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 $({\mathrm{\ell }}_p({\mathrm{\nabla }}^{(\gamma )}),\mathfrak{X}),$ where $\mathfrak{X}$ is any of the spaces ${\mathrm{\ell }}_{\mathrm{\infty }},$ $c,$ $c_0$ and ${\mathrm{\ell }}_1$.