Fractional q-calculus on a time scale

The study of fractional q-calculus in this paper serves as a bridge between the fractional q-calculus in the literature and the fractional q-calculus on a time scale T-t0 = {t : t = t(0)q(n), n a nonnegative integer} boolean OR {0}, where t(0) is an element of R and 0 < q < 1. By use of time scale calculus notation, we find the proof of many results more straight forward. We shall develop some properties of fractional q-calculus, we shall develop some properties of a q-Laplace transform, and then we shall employ the q-Laplace transform to solve fractional q-difference equations.