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 [Formula: see text] [Formula: see text] , in a neighborhood of the unique fixed point [Formula: see text] of the strictly increasing continuous function β, defined on an interval [Formula: see text] . These equations are based on the general quantum difference operator [Formula: see text] , which is defined by [Formula: see text] , [Formula: see text] . 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.