Sequence spaces constructed by using $\mathfrak{q}$-Pell-Lucas matrix and its geometric properties

Abstract

In this paper, the $\mathfrak{q}$-Pell-Lucas matrix, denoted by
$\grave{\text{Q}}(\mathfrak{q}) = (\grave{\text{Q}}_{{\mathrm{n}} {{\mathrm{k}}}}^{(\mathfrak{q})})_{{\mathrm{n}},{{\mathrm{k}}} \in \mathbb{N}_0}$ is defined by
\[
\grave{\text{Q}}_{{\mathrm{n}} {{\mathrm{k}}}}^{(\mathfrak{q})} =
\begin{cases}
\dfrac{{(\mathfrak{q}+1)}\grave{\text{Q}}_{{{\mathrm{k}}}}(\mathfrak{q})}{{(2+\mathfrak{q})}\grave{\text{Q}}_{{\mathrm{n}}}(\mathfrak{q})+{\mathfrak{q}}\grave{\text{Q}}_{{\mathrm{n}}-1}(\mathfrak{q})}, & 0 \leq {{\mathrm{k}}} \leq {\mathrm{n}}, \\
0, & {{\mathrm{k}}} > {\mathrm{n}},
\end{cases}
\]
and \(\{ \grave{\text{Q}}_{{\mathrm{k}}}{(\mathfrak{q})}\} \) corresponds to terms of the \( \mathfrak{q} \)-Pell-Lucas sequence. The sequence \(\{ \grave{\text{Q}}_{{\mathrm{k}}}{(\mathfrak{q} )}\}\) is defined by \[\grave{\text{Q}}_{{{\mathrm{k}}}}(\mathfrak{q})= \left\{2\grave{\text{Q}}_{{{\mathrm{k}}}-1}(\mathfrak{q} )+\mathfrak{q}^{{{}}}\grave{\text{Q}}_{{{\mathrm{k}}}-2}(\mathfrak{q} )\right\} \text{~for}~{{\mathrm{k}}} \leq 2,~ \grave{\text{Q}}_{0}(\mathfrak{q} )=2,\grave{\text{Q}}_{1}(\mathfrak{q} )=2.\]
The $\mathfrak{q}$-Pell-Lucas matrix serves as the foundation for the construction of matrix domains known as the $\mathfrak{q}$-Pell-Lucas sequence spaces. Within these spaces, we develop a Schauder basis, carry out a detailed investigation of operator ideals, and provide a comprehensive study of the geometric properties of $\ell_p(\grave{\text{Q}}(\mathfrak{q}))$ and $\ell_\infty(\grave{\text{Q}}(\mathfrak{q}))$, particularly addressing the Dunford--Pettis property, and the solidity property.}
\keywordstkjm{q-Pell-Lucas numbers, sequence space, Schauder basis, operator ideal, geometric property.