What question did this study set out to answer?

To introduce a type of generalized quaternions whose components are based on higher-order generalized Fibonacci hyper-numbers.

February 19, 2026

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

Key Points

To introduce a type of generalized quaternions whose components are based on higher-order generalized Fibonacci hyper-numbers.
Derived a Binet-like formula for the new generalized quaternions.
Established a recurrence relation related to these quaternions.
Created a generating function and an exponential generating function for calculations.
Explored summation properties of the generalized quaternions.
Formulated a new type of quaternions with specific mathematical properties.
Derived useful recurrence relations that help in further computations.
Provided generating functions that simplify the understanding of these quaternions.

Abstract

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.

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

Key Points

Abstract

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