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The paper presents a method of analysing a class of nonlinear time-varying discrete systems having polynomial-type nonlinearities. The analysis makes use of Volterra-series representation and multidimensional parametric z-transfer functions. The method is further extended to determine the output response between sampling instants through the use of multidimensional parametric modified z-transfer functions. The terminal-state theorems for multidimensional parametric z-transforms are presented. This new method of analysis is applied, for illustration, to a practical time-varying discrete system involving a nonlinear feedback element.