General theory of the higher-order quaternion linear difference equations via the complex adjoint matrix and the quaternion characteristic polynomial
Chao Wang, Desu Chen, Zhien Li
Abstract
Since the non-commutativity and particular structure of the quaternion algebra, the quternion difference equations (short for QDCEs) have a large difference from the classical theory of difference equations. In this paper, we establish a general theory of the higher-order linear QDCEs including the criteria of the linear independence (or dependence) of the discrete functions in the quaternion space, Liouville formulas, the structure theorems of the general solutions and the particular solutions with the quaternion power and trigonometric form, etc. By introducing the complex adjoint difference equations and the quaternion characteristic polynomial of the higher-order linear QDCEs, some basic results of the homogeneous and non-homogeneous difference equations are obtained. Through the analysis of the complex adjoint matrix and the quaternion eigenvalue, the general solutions of the higher-order linear QDCEs with variable and with constant coefficients are established. Several methods of obtaining the general solutions for the higher-order linear QDCEs are demonstrated and several examples are provided to illustrate the feasibility of our obtained results.