On homogeneous second order linear general quantum difference equations

In this paper, we prove the existence and uniqueness of solutions of the β-Cauchy problem of second order β-difference equations $$a_{0}(t)D_{\beta}^{2}y(t)+a_{1}(t)D_{\beta}y(t)+a_{2}(t)y(t)=b(t),\quad t \in I, $$ $a_{0}(t)\neq0$ , in a neighborhood of the unique fixed point $s_{0}$ of the strictly increasing continuous function β, defined on an interval $I\subseteq{\mathbb{R}}$ . These equations are based on the general quantum difference operator $D_{\beta}$ , which is defined by $D_{\beta}{f(t)}= (f(\beta(t))-f(t) )/ (\beta(t)-t )$ , $\beta(t)\neq t$ . We also construct a fundamental set of solutions for the second order linear homogeneous β-difference equations when the coefficients are constants and study the different cases of the roots of their characteristic equations. Finally, we drive the Euler-Cauchy β-difference equation.