A Note on Hyperbolic Generalized Pierre Numbers

Abstract

In this study, we present a new family of number sequences called the generalized hyperbolic Pierre numbers, defined within the bidimensional Clifford algebra of hyperbolic numbers. This algebraic framework enables the extension of classical sequence theory into the hypercomplex domain, offering both structural and analytical enrichment. As notable special cases, we examine the hyperbolic Pierre numbers and hyperbolic Pierre Lucas numbers, analyzing their algebraic properties, characteristic behaviors, and mutual relationships in detail. We derive closed-form expressions through Binet-type formulas, construct generating functions that reflect the recursive nature of the sequences, and establish summation identities that reveal deeper arithmetic patterns. Furthermore, we develop matrix representations for each sequence, providing a compact and elegant algebraic tool for modeling and manipulating their evolution. This research contributes to the broader theory of hypercomplex number sequences and proposes a novel approach to generalizing classical sequences within Clifford algebraic systems. The results have potential applications not only in pure mathematics but also in fields such as cryptography, numerical analysis, modeling of symmetric structures, solving differential equations, and algebraic representation of physical systems. In this context, the study lays a solid foundation for advanced investigations into the reinterpretation of number sequences in hyperbolic spaces.