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

In this study we develop a q-Fibonacci matrix \(\mathcal {F}(q)=(f^q_{nv})_{n,v\in \mathbb {N}_0}\) $\mathcal{F}(q)={(f_{\mathit{nv}}^q)}_{n,v\in {\mathbb{N}}_0}$ given by \(\begin{aligned} f^q_{nv}=\left\{ \begin{array}{ccc} q^{v+1}\frac{f_{v+1}(q)}{f_{n+3}(q)-1}&{}, &{} 0\le v\le n, \\ 0 &{}, &{} v>n. \end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill f_{\mathit{nv}}^q=\left\{\begin{array}{ccc}{q}^{v+1}\frac{f_{v+1}(q)}{f_{n+3}(q)-1}& ,& 0\le v\le n,\\ 0& ,& v>n.\end{array}\right)\end{array}$$ where \(\left( f_v(q)\right)\) $\left(f_v,(q)\right)$ represents a sequence of q-Fibonacci numbers. By utilizing the matrix \(\mathcal {F}(q)\) $\mathcal{F}(q)$ , we define matrix domains \(\ell _p (\mathcal {F}(q)):=(\ell _p)_{\mathcal {F}(q)}\) ${\ell }_p(\mathcal{F}(q)):={({\ell }_p)}_{\mathcal{F}(q)}$ \((0<p< \infty )\) $(0<p<\infty )$ and \(\ell _\infty (\mathcal {F}(q)):=(\ell _\infty )_{\mathcal {F}(q)}\) ${\ell }_{\infty }(\mathcal{F}(q)):={({\ell }_{\infty })}_{\mathcal{F}(q)}$ also known as q-Fibonacci sequence spaces. We obtain Schauder basis for the space \(\ell _p (\mathcal {F}(q))\) ${\ell }_p(\mathcal{F}(q))$ and determine Alpha-( \(\alpha\) $\alpha$ -), Beta-( \(\beta\) $\beta$ -) and Gamma-( \(\gamma\) $\gamma$ -) duals of the newly defined spaces. We obtain some results related to matrix transformations from the spaces \(\ell _p(\mathcal {F}(q))\) ${\ell }_p(\mathcal{F}(q))$ and \(\ell _\infty (\mathcal {F}(q))\) ${\ell }_{\infty }(\mathcal{F}(q))$ to classical sequence spaces \(\ell _\infty ,\) ${\ell }_{\infty },$ c and \(c_0\) $c_0$ . We also examined some of the geometric properties like approximation property, Dunford–Pettis property, Hahn–Banach extension property, and rotundity of the spaces \(\ell _p(\mathcal {F}(q))\) ${\ell }_p(\mathcal{F}(q))$ and \(\ell _\infty (\mathcal {F}(q))\) ${\ell }_{\infty }(\mathcal{F}(q))$ .