In this paper, we present a new definition, referred to as the Francois sequence,
related to the Lucas-like form of the Leonardo sequence. We also introduce the hyperbolic
Leonardo and hyperbolic Francois quaternions. Afterward, we derive the Binet-like formulas
and their generating functions. Moreover, we provide some binomial sums, Honsberger-like,
d’Ocagne-like, Catalan-like, and Cassini-like identities of the hyperbolic Leonardo quaternions
and hyperbolic Francois quaternions that allow an understanding of the quaternions’ proper-
ties and their relation to the Francois sequence and Leonardo sequence. Finally, considering
the results presented in this study, we discuss the need for further research in this field.