On Mersenne Finite Operators and Their Quaternion Counterparts

In this study, we introduce the Mersenne finite operator sequences, derived by applying a finite operator to Mersenne sequences, and investigate their fundamental properties. We establish the recurrence relations, derive a Binet-like formula, and present the generating functions associated with these sequences. Extending the analysis, we define the Mersenne finite operator quaternions and explore their structural properties, including recurrence relations, Binet-like representations, and generating functions. Special cases of these quaternions are examined, revealing connections with forward, backward, and mean difference operators. Moreover, we generalize well-known identities such as the Catalan, Cassini, and d’Ocagnes identities to the framework of Mersenne finite operator quaternions. The study further provides matrix representations of these quaternions, highlighting new relationships and structural insights. The obtained results contribute to the theoretical understanding of Mersenne-type sequences and their quaternion counterparts, offering potential applications in number theory and applied mathematics.