(untitled) (European Journal of Pure and Applied Mathematics, 2025)
In this paper, we investigate new identities involving generalized Fibonacci and Lucas sequences, as well as their associated quaternions. After establishing the fundamental properties of these generalized number sequences, we derive quaternionic extensions of product-difference identities originally introduced by Everman and Koshy for Fibonacci numbers. These results not only generalize classical identities, but also reveal new algebraic structures within the framework of generalized quaternion sequences.