Some results on generalized quaternions algebra with generalized Fibonacci quaternions

Recently, the most general form of the quaternion algebra on 3 parameters (3PGQ) has been introduced, which prompted us to look for some properties associated with this algebra, which will be called k λ 1 ,λ 2 ,λ 3 in this article. This paper consists of two main parts: the first part focuses on derivations, while the second part deals with the Fi-bonacci sequence, and the properties of both parts are related to the algebra k λ 1 ,λ 2 ,λ 3 . First we determine the derivations of the algebra k λ 1 ,λ 2 ,λ 3 and get the algebra Der( k λ 1 ,λ 2 ,λ 3 ) of derivations of k λ 1 ,λ 2 ,λ 3 . Then we were able to obtain generalized derivations that have been studied by mathematicians in the context of algebras of certain normed spaces, and semi-prime and prime rings. In the second part, we introduce some properties of the Fi-bonacci quaternions in the generalized quaternion algebra k λ 1 ,λ 2 ,λ 3 , which allow us to assume that the algebra k λ 1 ,λ 2 ,λ 3 is not always a division algebra.