On the Leonardo quaternions sequence

Authors : Patrícia Beites, Paula Maria Machado Cruz Catarino

Pages : 1001-1023

Doi:10.15672/hujms.1197693

View : 221 | Download : 324

Publication Date : 2024-08-27

Article Type : Research Paper

Abstract :In the present work, a new sequence of quaternions related to the Leonardo numbers -- named the Leonardo quaternions sequence -- is defined and studied. Binet\'s formula and certain sum and binomial-sum identities, some of which derived from the mentioned formula, are established. Tagiuri-Vajda\'s identity and, as consequences, Catalan\'s identity, d\'Ocagne\'s identity and Cassini\'s identity are presented. Furthermore, applying Catalan\'s identity, and the connection between composition algebras and vector cross product algebras, Gelin-Cesàro\'s identity is also stated and proved. Finally, the generating function, the exponential generating function and the Poisson generating function are deduced. In addition to the results on Leonardo quaternions, known results on Leonardo numbers and on Fibonacci quaternions are extended.
Keywords : Leonardo numbers, Leonardo quaternions, Binets formula, Tagiuri Vajdas, Sum and Binomial sum identities, generating function