| Summary: | Abstract 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 Dβ $D_{\beta }$ which is defined by Dβf(t)=(f(β(t))−f(t))/(β(t)−t) $D_{\beta }{f(t)}= (f(\beta (t))-f(t) )/ (\beta (t)-t )$, β(t)≠t $\beta (t)\neq t$, where β is a strictly increasing continuous function defined on an interval I⊆R $I\subseteq \mathbb{R}$ that has only one fixed point s0∈I $s_{0}\in {I}$. 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 Dβ $D_{\beta }$, 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.
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