MALAYA JOURNAL OF MATEMATIK Malaya J. Mat. 11(03)(2023), 324–331. http://doi.org/10.26637/mjm1103/009 The sequence of the hyperbolic k-Padovan quaternions RENATA VIEIRA * 1 ,FRANCISCO REGIS ALVES 2 AND PAULA CATARINO 3 1 Post-Graduate Program in Education of the Nordeste Education Network – Polo RENOEN-UFC, Federal University of Ceara, Brazil. 2 Federal Institute of Science and Technology Education of the State of Cear´ a, Brazil. 3 University of Tr´ as-os-Montes and Alto Douro, Portugal. Received 01 August 2022; Accepted 16 June 2023 Abstract. This work introduces the hyperbolic k-Padovan quaternion sequence, performing the process of complexification of linear and recurrent sequences, more specifically of the generalized Padovan sequence. In this sense, there is the study of some properties around this sequence, deepening the investigative mathematical study of these numbers. AMS Subject Classifications: 11B37, 11B39. Keywords: hyperbolic numbers, quaternions, k-Padovan sequence. Contents 1 Introduction and Background 324 2 The hyperbolic k-Padovan quaternions 325 3 Some properties 326 4 Conclusion 328 5 Acknowledgement 328 1. Introduction and Background Studies of recursive linear sequences have been noticed in the mathematical literature. Based on this, there is the concern to carry out an investigative study on the process of complexification of certain sequences. So soon, in this work, the hyperbolic quaternion k-Padovan sequence is introduced, presenting algebraic properties around these numbers. The Padovan sequence is a linear and recurrent third-order sequence, named after the Italian architect Richard Padovan. Thus, its recurrence is given by: P n = P n−2 + P n−3 ,n ≥ 3 and being P 0 = P 1 = P 2 =1 your initial conditions [13–16]. The quaternions were developed by Willian Rowan Hamilton (1805-1865), arose from the attempt to generalize complex numbers in the form z = a + bi in three dimensions [10]. Thus are presented as formal sums of scalars with usual vectors of three-dimensional space, existing four dimensions. Second Halici (2012) [8], a quaternion is a hyper-complex number and is described by: q = a + bi + cj + dk ∗ Corresponding author. Email address: re.passosm@gmail.com (Renata Vieira) https://www.malayajournal.org/index.php/mjm/index ©2023 by the authors.