Discrete Calculus

Firstly, we define the discrete differentiation and integration of functions and their arithmetic properties. By using these properties, we can easily calculate many important summation formulas in particular a closed formula for the sum of kth powers of the first n consecutive positive integers, an expression for the discrete gamma function, as well as a formula for the number of regions in which x hyperspaces in general position, divide the n-dimensional space Rn. It turns out that this number can be given as a function of x expressible as a Taylor expansion at the origin. Secondly, by introducing an analog of the exponential function, we treat the difference equations as a discrete differential equations. For example, we solve the arithmetic progression as a first order and Fibonacci sequence as a second order Cauchy discrete differential equation problems. Finally, in the remaining three sections, we deal with the methods of the variation of parameters, the solution of the Cauchy problem for non-homogeneous differential equations, as well as the boundary value problem both for homogeneous and non-homogeneous differential equations of second order in discrete cases.