Some q-fractional order difference sequence spaces

Abstract

This paper intends to develop a $q$-difference operator ${\nabla }_q^{({\hat{I}}^3)}$ of fractional order ${\hat{I}}^3$, and give several intriguing properties of this new difference operator. Our main focus remains on the construction of sequence spaces ${\ell }_p({\nabla }^{({\hat{I}}^3)})$ and ${\ell }_{\infty }({\nabla }^{({\hat{I}}^3)})$, at the same time comparing these spaces with those already exist in the literature. Apart from obtaining Schauder basis, we determine $\hat{I}\pm$-, ${\hat{I}}^2$-, and ${\hat{I}}^3$-duals of the newly defined spaces. A section is also devoted for characterizing matrix classes $({\ell }_p({\nabla }^{({\hat{I}}^3)}),\mathfrak{X}),$ where $\mathfrak{X}$ is any of the spaces ${\ell }_{\infty },$ $c,$ $c_0$ and ${\ell }_1$.

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