This article proposes a new strategy to deal with linear parameter-varying discrete-time systems, whose time-varying parameters can be written as solutions (such as exponential, trigonometric, or periodic function) of a linear difference equation (DE). The novelty is to explicitly exploit the precise knowledge of the function describing the time-varying parameter by incorporating the associated DE in the conditions, providing less conservative results when compared with conventional approaches based on bounded or arbitrary rates of variation. The advantage of the method comes from the fact that, differently from the available methods, the pointwise stability for the whole domain of the time-varying parameters is not a necessary condition to obtain feasible solutions. The applicability and benefits of the proposed technique are investigated in terms of numerical examples concerning robust stability analysis, H-infinity filtering, and H-infinity state-feedback control. As a final contribution, the problem of time-varying sampling periods in the context of networked control systems is investigated using the proposed strategy. A numerical example based on a practical application is presented to illustrate the superiority of the approach when compared to methods from the literature based on matrix exponential computation.