Calculus of finite differences

Summary

The calculus of finite differences studies functions and sequences through their finite differences and sums. It does so in the way that ordinary calculus uses derivatives and integrals. Its main tools are the forward difference operator $\Delta$, the shift operator $E = I + \Delta$, and the indefinite sum, which inverts $\Delta$. Many rules of calculus have exact discrete counterparts, such as summation by parts and the fundamental theorem of the calculus of finite differences. It is the basis of the theory of difference equations, of interpolation at equally spaced points, and of closed forms for finite sums.