On Generalized Quaternions Whose Components are Higher‐Order Generalized Fibonacci Hyper‐Numbers With q$$ q $$‐Integer Numbers

ABSTRACT The main aim of this paper is to introduce generalized quaternions whose components are higher‐order generalized Fibonacci hyper‐numbers with ‐integer numbers. We first derive the Binet‐like formula for this newly established type of generalized quaternions. With the help of the Binet‐like formula, we give recurrence relation, generating function, exponential generating function, and some useful summation properties of these generalized quaternions.