On some new difference sequence spaces of fractional order

TL;DR

This paper introduces new fractional difference sequence spaces, analyzes their duals and bases, and explores their class properties, expanding the mathematical framework of fractional difference operators.

Contribution

The paper defines novel fractional difference sequence spaces and determines their duals and bases, providing new tools for sequence space analysis.

Findings

Defined new sequence spaces $c_0( abla^{(eta)},u)$ and $c( abla^{(eta)},u)$.

Calculated the $eta$-duals and Schauder bases of these spaces.

Explored classes $( u( abla^{(eta)},u): ho)$ for various sequence spaces.

Abstract

Let $\Delta^{(\alpha)}$ denote the fractional difference operator. In this paper, we define new difference sequence spaces $c_0(\Gamma,\Delta^{(\alpha)},u)$ and $c(\Gamma,\Delta^{(\alpha)},u)$. Also, the $\beta-$ dual of the spaces $c_0(\Gamma,\Delta^{(\alpha)},u)$ and $c(\Gamma,\Delta^{(\alpha)},u)$ are determined and calculated their Schauder basis. Furthermore, the classes $(\mu(\Gamma,\Delta^{(\alpha)},u):\lambda)$ where $\mu\in \{c_{0},c\}$ and $\lambda \in \{c_{0},c,\ell_{\infty},\ell_1\}$ .