Gaussian Pell and Gaussian Pell-Lucas quaternions

Abstract

The main aim of this work is to introduce the Gaussian Pell quaternion QGpn and Gaussian Pell-Lucas quaternion QGqn, where the components of QGpn and QGqn are Pell numbers pn and Pell-Lucas numbers qn, respectively. Firstly, we obtain the recurrence relations and Binet formulas for QGpn and QGqn. We use Binet formulas to prove Cassini?s identity for these quaternions. Furthermore, we give some basic identities for QGpn and QGqn such as some summation formulas, the terms with negative indices and the generating functions for these complex quaternions.