On homogeneous second order linear general quantum difference equations

In this paper, we prove the existence and uniqueness of solutions of the beta-Cauchy problem of second order beta-difference equations a(0)(t) D(beta)(2)y(t) + a(1)(t) D(beta)y(t) + a(2)(t) y(t) = b(t), t is an element of I, a(0)(t) not equal 0, in a neighborhood of the unique fixed point s(0) of the strictly increasing continuous function beta, defined on an interval I subset of 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) beta not equal t. We also construct a fundamental set of solutions for the second order linear homogeneous beta-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 beta-difference equation.