An Extension of Maddox’s Paranormed Sequence Spaces with Narayana Numbers

This study focuses on constructing an extended version of Maddox’s paranormed sequence spaces, denoted by $c_0(N, p)$, $c(N, p)$, $\ell_\infty(N, p)$, and $\ell(N, p)$. The study aims to define and investigate the characteristics of these sequence spaces, along with their paranormed extensions. In particular, it develops a theoretical framework for Narayana sequence spaces, establishing their topological, algebraic, and matrix transformation properties. The Schauder basis for these spaces is introduced, laying the foundation for further functional analysis. The research also examines the $alpha$-, $\beta$-, and $\gamma$- duals of these spaces and investigates the conditions under which matrix transformations preserve their structural properties. The findings highlight the equivalences between these spaces and classical sequence spaces such as $\ell_1$, $c$, and $\ell_\infty$.