Exploring the $ q $-analogue of Fibonacci sequence spaces associated with $ c $ and $ c_0 $

Abstract

We have proposed a $ q $-analogue $ c(\mathcal{F}(q)) $ and $ c_0(\mathcal{F}(q)) $ of Fibonacci sequence spaces, where $\mathcal{F}(q) = (f^q_{km})$ denotes a $ q $-Fibonacci matrix defined in the following manner: \begin{document}$ f^q_{km} = \begin{cases} q^{m+1} \frac{f_{m+1}(q)}{f_{k+3}(q) - 1}, & \text{if } 0 \leq m \leq k, \\ 0, & \text{if } m > k, \end{cases} $\end{document} for all $ k, m \in \mathbb{Z}^+_0 $, where $(f_k(q))$ denotes a sequence of $ q $-Fibonacci numbers. We developed a Schauder basis and determined several important duals ($ \alpha $-, $ \beta $-, $ \gamma $-) of the aforesaid constructed spaces $ c(\mathcal{F}(q)) $ and $ c_0(\mathcal{F}(q)) $. Additionally, we examined certain characterization results for the matrix class $(\mathfrak{U}, \mathfrak{V})$, where $\mathfrak{U} \in \{c(\mathcal{F}(q)), c_0(\mathcal{F}(q))\}$ and $\mathfrak{V} \in \{\ell_{\infty}, c, c_0, \ell_1\}$. Essential conditions for the compactness of the matrix operators on the space $ c_0(\mathcal{F}(q)) $ via the Hausdorff measure of noncompactness (Hmnc) were presented.

Keywords

• $ q $-fibonacci sequence spaces
• duals
• matrix mappings
• compactness