The transition from a known Taylor series of a known function f(x) to a new function primarily defined by the infinite power series with coefficients f(n)(0) from the Taylor series of the function f(x) can be made by an integral transformation which is a modified Laplace transformation and is called Sumudu transformation. It makes the transition from the Exponential series to the Geometric series and may help to evaluate new infinite power series from known Taylor series. The Sumudu transformation is demonstrated to be a limiting case of Fractional integration. Apart from the basic Sumudu integral transformation we discuss a modification where the coefficients from the Taylor series are not changed to f(n)(0) but only to . Beside simple examples our applications are mainly concerned to calculate new Generating functions for Hermite polynomials from the basic ones.