Based on the parametric characteristic of the nth-order generalized frequency response function (GFRF) for non-linear systems described by a non-linear differential equation (NDE) model, a mapping function from the parametric characteristics to the GFRFs is established, by which the nth-order GFRF can be directly written into a more straightforward and meaningful form in terms of the first order GFRF, i.e., an n-degree polynomial function of the first order GFRF. The new expression has no recursive relationship between different order GFRFs, and demonstrates some new properties of the GFRFs which can explicitly unveil the linear and non-linear factors included in the GFRFs, and reveal clearly the relationship between the nth-order GFRF and its parametric characteristic, as well as the relationship between the nth-order GFRF and the first order GFRF. The new results provide a novel and useful insight into the frequency domain analysis and design of non-linear systems based on the GFRFs. Several examples are given to illustrate the theoretical results.