On the Quaternion Padovan Numbers

Dişkaya, Orhan | Menken, Hamza

In the present work, we explore the concept of quaternion Padovan sequences, extending the classical Padovan sequence into the realm of quaternions. Quaternions, introduced by Hamilton in 1843, are a number system that generalizes complex numbers into four dimensions, with applications in various fields such as quantum physics, computer graphics, and three-dimensional rotations. The study begins by reviewing the definition and properties of quaternions, including their non- commutative multiplication rules and the formulation of quaternion numbers. The Padovan sequence is defined by the recurrence relation Pn+3 = Pn+1 + Pn with initial conditions P0 = P1 = P2 =1. This sequence is generalized to quaternions by defining quaternion Padovan numbers, denoted QP (a,b, c, d) , wh...