New Triangular q−Fibonacci Matrix

Abstract

In this study, we construct a new triangular $q-$analogue of the $q-$Fibonacci matrix $\widetilde{f}_{q}=\Big({f}_{nk}(q)\Big)$ defined by

\begin{align*}
{f}_{nk}(q)=\left\{ \begin{array}{lll} \dfrac{q^{k}f_{k}(q)}{f_{n+2}(q)-1} &, 1\leq k\leq n\\
0 &, \textrm{otherwise} \end{array} \right..
\end{align*}

After, we use this analogue to define the sequence spaces $c(\widetilde{f}_{q})$, $c_{0}(\widetilde{f}_{q})$, $\ell_{\infty}(\widetilde{f}_{q})$, $\ell_{p}(\widetilde{f}_{q})$ $\left(1\leq p< \infty\right)$. Then, we provide some inclusion relations for these spaces and examine a few topological characteristics. Furthermore, we construct a basis for the space $\ell_{p}(\widetilde{f}_{q})$, calculate $\alpha-$, $\beta-$, $\gamma-$duals of the same space, describe certain matrix classes, and look at some geometric properties.