The Fibonacci and Lucas Generalized Quaternionic Sequences Over HGC Numbers

Abstract

In this paper, with the use of generalized complex and hyperbolic numbers, we build the theory of generalized quaternions with hyperbolic-generalized complex (HGC) numbers as coefficients. Additionally, certain associated theoretical universal results involving HGC Fibonacci and Lucas numbers, including their generalized quaternions, are established. With this approach, bihyperbolic, hyperbolic-complex, and hyperbolic-dual generalized quaternions can be determined for specified values of p is an element of R. It is also possible to study numerous types of quaternions with HGC number coefficients and their attributes depending on the choice of the real values alpha and beta.