A Novel Study on q-Fibonacci Sequence Spaces and Their Geometric Properties

Abstract

In this study we develop a q-Fibonacci matrix \(\mathcal {F}(q)=(f^q_{nv})_{n,v\in \mathbb {N}_0}\) given by

$$\begin{aligned} f^q_{nv}=\left\{ \begin{array}{ccc} q^{v+1}\frac{f_{v+1}(q)}{f_{n+3}(q)-1}&{}, &{} 0\le v\le n, \\ 0 &{}, &{} v>n. \end{array}\right. \end{aligned}$$

where \(\left( f_v(q)\right)\) represents a sequence of q-Fibonacci numbers. By utilizing the matrix \(\mathcal {F}(q)\), we define matrix domains \(\ell _p (\mathcal {F}(q)):=(\ell _p)_{\mathcal {F}(q)}\) \((0<p< \infty )\) and \(\ell _\infty (\mathcal {F}(q)):=(\ell _\infty )_{\mathcal {F}(q)}\) also known as q-Fibonacci sequence spaces. We obtain Schauder basis for the space \(\ell _p (\mathcal {F}(q))\) and determine Alpha-(\(\alpha\)-), Beta-(\(\beta\)-) and Gamma-(\(\gamma\)-) duals of the newly defined spaces. We obtain some results related to matrix transformations from the spaces \(\ell _p(\mathcal {F}(q))\) and \(\ell _\infty (\mathcal {F}(q))\) to classical sequence spaces \(\ell _\infty ,\) c and \(c_0\). We also examined some of the geometric properties like approximation property, Dunford–Pettis property, Hahn–Banach extension property, and rotundity of the spaces \(\ell _p(\mathcal {F}(q))\) and \(\ell _\infty (\mathcal {F}(q))\).