The monograph presented shows new models of mathematical uniform difference operators and their applications for the analytical sum determination of infinite series. Various properties of the uniform difference operator and its inverse form are presented here. The domain of the uniform difference operator is the discrete number space. The monograph takes the further application steps of the uniform difference operator and presents the general and particular solutions of n-order difference and recurrence linear homogeneous and non-homogeneous ordinary and partial equations with constant and variable coefficients. The monograph presents numerous calculations and solutions of the examples of choices. In this monograph the mathematical and numerical achievements are presented and discussed especially solutions of summation and recurrence equations, applied in the computational mechanics computer science and control systems.