Sequence Spaces and Operator Ideals induced by the q-Bronze Leonardo-Lucas Matrix

Abstract

This paper introduces a \( q \)-Bronze Leonardo-Lucas matrix \( \aleph(q) = ({\breve{\zeta}}_{nk}^{(q)})_{n, k \in \mathbb{N}} \), where the elements are defined by
\[
\breve{\zeta}_{nk}^{(q)} =
\begin{cases}
\dfrac{3~q^{k-1}~\breve{\zeta}_{k}(q)}{4\breve{\zeta}_n(q)+\breve{\zeta}_{n-1}(q)+3n-10}, & 1 \leq k \leq n, \\\\
0, & k > n,
\end{cases}
\]
with \( \{\breve{\zeta}_{n}\} \) representing the \( q \)-Bronze Leonardo-Lucas sequence. Using \( \aleph(q) \) is define by
\[\breve{\zeta}_n(q)=(2+q^{k-1})\breve{\zeta}_{n-1}(q)+q^{n-1}\breve{\zeta}_{n-2}(q)-3\]
for \(n \geq 2\),~\(\breve{\zeta}_0(q)=3,\breve{\zeta}_1(q)=4\). We introduce the matrix domains \( \ell_p(\aleph(q)) = (\ell_p)_{\aleph(q)} \) for \(1 \leq p < \infty\), along with \( \ell_\infty(\aleph(q)) = (\ell_\infty)_{\aleph(q)} \), \( c_0(\aleph(q)) = (c_0)_{\aleph(q)} \), and \( c(\aleph(q)) = (c)_{\aleph(q)} \), calling them the \( q \)-Bronze Leonardo-Lucas sequence spaces. In this context, we derive the Schauder basis for the space \( \ell_p(\aleph(q)) \) for \( 1 \leq p < \infty \). %and determine the \(\alpha-\), \(\beta-\), and \(\aleph(q)-\) duals for the newly defined spaces.
We establish several results concerning operator ideals associated with these newly defined sequence spaces.
Additionally, we explore various geometric characteristics of \( \ell_p(\aleph(q)) \) and \( \ell_\infty(\aleph(q)) \) and lastly, we analyze the solidity characteristics of these sequence spaces.