Unrestricted Fibonacci and Lucas quaternions

Abstract

Many quaternion numbers associated with Fibonacci and Lucas numbers or even their generalizations have been defined and widely discussed so far. In all the studies, the coefficients of these quaternions have been selected from consecutive terms of these numbers. In this study, we define other generalizations for the usual Fibonacci and Lucas quaternions. We also present some properties, including the Binet's formulas and d'Ocagne's identities, for these types of quaternions.