On the Solutions of Quaternion Difference Equations in Terms of Generalized Fibonacci-Type Numbers

The aim of this paper is to investigate the solution following difference equation zn+1=(pn)−1,n∈N0,N0=N∪0 where pn=a+bzn+czn−1zn with parameters a, b, c and initial values z−1,z0 are nonzero quaternions such that their solutions associated generalized Fibonacci-type numbers. Furthermore, we deal symmetric system equations given by zn+1=(qn)−1,wn+1=(rn)−1,n∈N0 qn=a+bwn+czn−1wn rn=a+bzn+cwn−1zn. We provide third-order quaternion linear in terms zeros characteristic polynomial equation.