Gaussian Pell and Gaussian Pell-Lucas Quaternions

Abstract

The main aim of this work is to introduce the Gaussian Pell quaternion $QGp_{n}$ and Gaussian Pell-Lucas quaternion $QGq_{n}$, where the components of $QGp_{n}$ and  $QGq_{n}$ are Pell numbers $p_{n}$ and Pell-Lucas numbers $q_{n}$, respectively. Firstly, we obtain the recurrence relations and Binet formulas for $QGp_{n}$ and  $QGq_{n}$. We use Binet formulas to prove Cassini's identity for these quaternions. Furthermore, we give some basic identities for $QGp_{n}$ and  $QGq_{n}$ such as some summation formulas, the terms with negative indices and the generating functions for these complex quaternions.