On the Domain of 4-Dimensional Catalan Matrix in the Space of Absolutely Summable Double Sequences

The primary objective of this study is to construct a novel double sequence space by utilizing the domain of a 4-dimensional (4D) matrix in the space $\mathcal{L}_u$ of the absolutely summable double sequences defined via the well-known Catalan numbers. Within this framework, various algebraic and topological properties of the newly introduced space are investigated. The study further aims to compute the $\alpha$-, $\beta(bp)$-, and $\gamma$-duals of the space. Another essential part of the research involves characterizing certain classes of matrix transformations from the newly defined space to some classical double sequence spaces and vice versa. These contributions are expected to enrich the theory of sequence spaces and matrix transformations by introducing new insights based on special integer sequences.