General theory of the higher-order linear quaternion q-difference equations through the quaternion determinant algorithm and the characteristic polynomial

Abstract

The general theory of quaternion q-difference equations is completely different from traditional q-difference equations in the complex space for the special algebraic structure of the quaternion space, such as the non-commutativity of multiplication among elements. In this paper, some properties of basic quaternion q-discrete functions, such as the quaternion q-exponential function, are obtained. The existence, uniqueness, and extension theorems of the solution for higher-order linear quaternion q-difference equations (QQDCEs) are established through constructing quaternion q-stepwise approximation sequences, and the equivalent variable transform between higher-order linear QQDCEs and quaternion linear q-difference equations is given. Moreover, some basic results, such as the Wronskian formula, Liouville formula, and general solution structure theorems of higher-order linear QQDCEs with constant and variable coefficients, are established by applying the quaternion characteristic polynomial and the quaternion determinant algorithm. In addition, some particular and general solution formulas of non-homogeneous QQDCEs are obtained under the non-commutative condition of quaternion elements. Finally, several examples are provided to illustrate the feasibility of our obtained results.