On homogeneous second order linear general quantum difference equations.

In this paper, we prove the existence and uniqueness of solutions of the ß-Cauchy problem of second order ß-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 ß, 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 ß-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 ß-difference equation.