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

Abstract

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

Keywords

• quaternion difference equation
• solution of difference equation
• recurrence sequence