On higher-order Gaussian Fibonacci quaternions and their properties

This paper ensures an extensive survey of the generalization of the Gauss Fibonacci quaternions especially as part of its enhancing importance in the disciplines of mathematics and physics. The main objectives of our work is to define and study the higher-order Gaussian Fibonacci quaternion $GFq_{n^{(r)}}$, where the components of $GFq_{n^{(r)}}$ are higher-order Gaussian Fibonacci numbers $GFq_{n^{(r)}}$. Initially, we obtain the Binet-like formula, some identities for $GFq_{n^{(r)}}$, the recursive relation and the summation formulas for $GFq_{n^{(r)}}$. We have derived the terms with negative indices of $GFq_{n^{(r)}}$, the generating functions, and exponential generating function for higher-order Gaussian Fibonacci quaternions $GFq_{n^{(r)}}$. Additionally, we have obtained some new identities for these types of quaternions by using the special matrices.