Paranormed Motzkin Difference Sequence Spaces of Fractional

This study presents a theoretical investigation of a newly defined sequence space, termed the “Motzkin fractional difference sequence space,” which is constructed through the combination of the Motzkin matrix derived from Motzkin numbers and fractional difference operators. Motzkin numbers, notable for their special role in discrete mathematics and combinatorial structures, are revisited in this work from both algebraic and topological perspectives. The completeness, normability, and topological properties of the defined spaces are thoroughly analyzed, and it is demonstrated that they are linearly isomorphic to classical sequence spaces via modulus functions. For each space, original Schauder bases are constructed, and the structures of normed Banach and quasi-paranormed spaces are rigorously established. Furthermore, the characterization of dual spaces is provided, relevant matrix transformations are identified, and norm estimations are presented. Algebraic analyses of Motzkin numbers contribute to numerical solutions, while their connections to symmetric structures offer new insights within the framework of operator theory. These findings also open pathways for interdisciplinary applications in computer science, biological modeling, and the analysis of physical systems.In conclusion, this study reveals that the newly defined sequence space based on Motzkin numbers offers original contributions to both theoretical and applied mathematics. By integrating the analysis of algebraic structures with the characterization of functional spaces, it provides a valuable addition to the existing literature.