A new approach to calculating the dynamics and equilibrium thermodynamics of an arbitrary (quantum or stochastic) system is presented. Its key points are representing the full propagator as a product of the harmonic-oscillator propagator with the configuration function, and expanding the configuration function (its exponent) in a power series in a given function of t. Recursion relations are obtained for the expansion coefficients which can be analytically evaluated for any number of degrees of freedom. This representation is particularly attractive for two reasons. Being structurally similar to the standard Taylorlike expansions for the propagator already known in the literature, it nevertheless shows a dramatic improvement over the latter in that it converges significantly better over a much broader range of t. Another attractive feature of the present expansion is that it is amenable to subsequent approximations. With this technique a minimal computational effort is required for constructing an improved global approximation for the propagator which is exact not only if t goes to zero, but also in the limit t-->infinity. Numerical applications to the coordinate space density matrix, quantum-mechanical time correlation function, and Fokker-Planck conditional probability show an accurate description of dynamical (statistical) properties to be already achieved for arbitrarily large times (small temperatures) with just the first term of the present expansion taken into account. Its use in a path integral means that a dramatic reduction of the number of integration variables which is required for convergence will be achieved even though simulations over very long times are desirable.