Theory of nth-order linear general quantum difference equations

In this paper, we derive the solutions of homogeneous and non-homogeneous nth-order linear general quantum difference equations based on the general quantum difference operator $D_{\beta }$ which is defined by $D_{\beta }{f(t)}= (f(\beta (t))-f(t) )/ (\beta (t)-t )$ , $\beta (t)\neq t$ , where β is a strictly increasing continuous function defined on an interval $I\subseteq \mathbb{R}$ that has only one fixed point $s_{0}\in {I}$ . We also give the sufficient conditions for the existence and uniqueness of solutions of the β-Cauchy problem of these equations. Furthermore, we present the fundamental set of solutions when the coefficients are constants, the β-Wronskian associated with $D_{\beta }$ , and Liouville’s formula for the β-difference equations. Finally, we introduce the undetermined coefficients, the variation of parameters, and the annihilator methods for the non-homogeneous β-difference equations.