Suleyman Aydinyuz^1,, Mustafa Asci¹

In this study, we define a new interesting generalization of quaternions called as generalized k-order Fibonacci and Lucas quaternions. We give some important results with specific choices. Depending on the d_i and q choices, we obtain k-order Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas quaternions. For k=2, we obtain the recurrence relations of known special numbers such as Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas quaternions. By multiplying the choices we made, we can obtain the quaternion definitions for other special numbers. We give generating functions for these quaternions. Also, we identify and prove the matrix representations for generalized k-order Fibonacci and Lucas quaternions. In this way, we obtain the matrix representations for usual Fibonacci, Lucas, Pell and the other special numbers known with the d_i and q values we chose and give some properties about matrix representations for generalized k-order Fibonacci and Lucas quaternions.

Fibonacci Numbers, Lucas Numbers, Generalized k-Order Fibonacci and Lucas Numbers, Quaternions, Generalized k-Order Fibonacci and Lucas Quaternions, Matrix Representations