Some q-fractional order difference sequence spaces

TL;DR

This paper introduces a new q-fractional difference operator and constructs associated sequence spaces, analyzing their properties, duals, and matrix mappings to expand the mathematical framework of fractional difference sequences.

Contribution

It develops a novel q-fractional difference operator and constructs new sequence spaces, providing their properties, duals, and matrix class characterizations.

Findings

Defined a new q-fractional difference operator $ abla^{( extgamma)}_q$.

Constructed and analyzed the properties of sequence spaces $ extell_p( abla^{( extgamma)})$ and $ extell_ extinfty( abla^{( extgamma)})$.

Determined dual spaces and characterized matrix classes involving these new sequence spaces.

Abstract

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 $\ell_1$.