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.

由于四元数代数的非交换性和特殊结构，四元数差分方程（QDCEs的简称）与经典的差分方程理论有很大的不同。在本文中，我们建立了高阶线性QDCE的一般理论，包括四元数空间中离散函数的线性独立性（或相关性）判据、Liouville公式、通解和特解的结构定理通过引入复伴随差分方程和高阶线性QDCE的四元数特征多项式，得到齐次和非齐次差分方程的一些基本结果。通过对复伴随矩阵和四元数特征值的分析，建立了变系数和常系数高阶线性QDCE的通解。演示了获得高阶线性 QDCE 通用解的几种方法，并提供了几个例子来说明我们获得的结果的可行性。