The Fibonacci and Lucas generalized quaternionic sequences over $\mathcal{HGC}$ numbers

In this paper, with the use of generalized complex and hyperbolic numbers, we build the theory of generalized quaternions with hyperbolic-generalized complex $(\mathcal{HGC})$ numbers as coefficients. Additionally, certain associated theoretical universal results involving $\mathcal{HGC}$ 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 $\mathfrak{p}\in\mathbb{R}$. It is also possible to study numerous types of quaternions with $\mathcal{HGC}$ number coefficients and their attributes depending on the choice of the real values and $\alpha$ and $\beta$.