Exploring the Hidden Symmetry of Quaternions: Dual Balancing and Cobalancing Numbers and Their Corresponding Quaternions

Abstract

This paper unveils the hidden symmetry of quaternions through the introduction of Dual Balancing and Dual Cobalancing numbers, along with their corresponding quaternions. We provide Binet's formulas, generating functions, and a multitude of intriguing properties to foster a comprehensive understanding of these concepts. Furthermore, we present matrix representations to offer a fresh perspective on the Dual Balancing and Dual Cobalancing quaternions. The implications of this study go beyond theoretical applications and may have practical implications in diverse fields.