Quaternion Fractional Difference with Quaternionic Fractional Order and Applications to Fractional Difference Equation

In this paper, we introduce the basic notions of the fractional summation, difference and q-difference with the quaternionic fractional order for the quaternion-valued functions and establish some of their basic properties. Based on this, the summation representations of solutions for the nonlinear quaternion-valued fractional difference equation and q-difference equation are obtained. In addition, several examples are provided to illustrate the feasibility of our obtained results in each section.