Some q-fractional order difference sequence spaces

TL;DR

This work extends $q$-calculus to a fractional order setting by defining the $q$-fractional difference operator $\nabla^{(\gamma)}_q$ via a $q$-gamma function expansion and representing it as a lower-triangular matrix. It introduces the sequence spaces $\ell_p(\nabla^{(\gamma)}_q)$ and $\ell_\infty(\nabla^{(\gamma)}_q)$ as domains of $\nabla^{(\gamma)}_q$, and analyzes their basic structure, including a Schauder basis for $\ell_p(\nabla^{(\gamma)}_q)$ and the lack of one for $\ell_\infty(\nabla^{(\gamma)}_q)$. The paper then determines the $\alpha$-, $\beta$-, and $\gamma$-duals of these spaces and develops a framework for matrix transformations between them and classical sequence spaces, detailing necessary and sufficient conditions via transformed coefficient matrices. Overall, the results generalize known difference spaces to the fractional $q$-setting and establish foundational duality and operator-transformation theory with potential applications in spectral analysis and functional analysis on $q$-difference spaces.

Abstract

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

Theorems & Definitions (30)

• Definition 1.1
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Lemma 2.1
• proof
• Theorem 2.3
• proof
• Theorem 2.4