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In this paper, we define split quaternions with components including Pell and Pell-Lucas number sequences. By using Binet's formulas and Cayley-Dickson's notation we introduce a new polar representation for split quaternions. This alternative representation, based on two complex number sequences, provides a new perspective on the structure of Pell and Pell-Lucas split quaternions and give a deeper understanding of their geometric interpretations and transformations. Furthermore, some fundamental properties and identities for these type of Pell and Pell-Lucas split quaternions are studied. In further the current paper, it would be valuable to replicate similar approaches polar representationin with Pell and Pell-Lucas Split quaternions.
Babadağ et al. (2024) studied this question.