On Dual Hyperbolic Narayana and Narayana-Lucas Hybrid Quaternions

In this present communication, we introduce the novel concepts of dual hyperbolic Narayana quaternions and dual hyperbolic Narayana-Lucas quaternions within the framework of hybrid numbers. We also explore the connections between these newly defined quaternions and examine the mathematical properties they share. Additionally, we find the recurrence relations, Binet formulas, generating functions, exponential generating functions, and other meaningful identities. These newly introduced quaternions have significant applications in the fields of quantum physics, computer graphics, and robotics. Additionally, these identities and relationships we established also play an important role in the field of number theory and combinatorics.