Three-Parameter Generalized Quaternions Based on Gaussian-Leonardo Numbers

This paper aims to extend the theory of quaternions by introducing a novel type of 3-parameter generalized quaternion, whose components are Gaussian Leonardo (abbreviated as G-L) numbers, referred to as the G-L 3-parameter generalized quaternions.Through systematic investigation of their recurrence relations, Binetlike formulas, generating functions, exponential generating functions, Poisson generating functions, as well as several important identities and summation formulas, we not only uncover the rich mathematical structure of these quaternions but also provide new tools for applications in related fields.Additionally, by utilizing the S-matrix and representation matrix of the G-L 3-parameter generalized quaternions, a new identity is discovered, further perfecting the theoretical framework.These results offer significant references for the deepening and expansion of quaternion theory.