k-ORDER FIBONACCI QUATERNIONS

Abstract

In this paper, we define and study another interesting generalization of the Fibonacci quaternions is called k-order Fibonacci quaternions. Then we obtain for k = 2 Fibonacci quaternions, for k = 3 Tribonacci quaternions and for k = 4 Tetranacci quaternions. We give generating function, the summation formula and some properties about k-order Fibonacci quaternions. Also, we identify and prove the matrix representation for korder Fibonacci quaternions. The Q(k) matrix given for k-order Fibonacci numbers is defined for k-order Fibonacci quaternions and after the matrices with the k-order Fibonacci quaternions is obtained with help of auxiliary matrices, important relationships and identities are established.