New results on generalized quaternions for hyper-numbers with Fibonacci and Lucas numbers components

This paper presents a comprehensive survey of the generalized quaternions of the hyper-numbers with Fibonacci and Lucas numbers components, particularly in light of their increasing significance in the fields of mathematics and physics. The main objectives of our paper are to introduce and examine the generalized quaternions for hyper-numbers with Fibonacci and Lucas numbers components, respectively. After that, we give Binet-like formulas, generating functions, exponential generating functions, and some summation formulas for these types of quaternions. Because of the hyper-numbers we have considered, the quaternion families we have defined are very general.