Dual Hyperbolic Generalized Adrien Numbers

This study introduces the generalized dual hyperbolic Adrien numbers, a novel extension of the classic Adrien framework, enriched by dual and hyperbolic algebraic structures. These sequences are constructed within a fourth-order linear recurrence system, offering intricate mathematical behavior and promising structural versatility. Special attention is devoted to two distinguished cases: the dual hyperbolic Adrien and dual hyperbolic Adrien–Lucas numbers, each revealing unique interrelations between dual numbers, hyperbolic units, and classical integer sequences. For each class, explicit Binet-type expressions are derived, enabling direct computation and closed-form analysis. Ordinary and exponential generating functions are presented to encapsulate the sequences’ evolution and provide analytical tools for combinatorial and algebraic exploration. Summation formulas are established to link consecutive terms and identify structural patterns across the sequences. Matrix representations are also constructed, encoding the recurrence relations and offering compact formulations suitable for computational applications. The proposed families of numbers serve not only as theoretical constructs but also as candidates for deeper investigations into symbolic computation, algebraic identities, and discrete dynamical models. Their formulation opens doors for applications in areas such as cryptography, quantum computation, and signal processing, where dual and hyperbolic systems are gaining renewed interest.