Indefinite sum

In discrete calculus the indefinite sum operator (also known as the antidifference operator), denoted by ${\textstyle \sum _{x}}$ or ${\displaystyle \Delta ^{-1}}$, is the linear operator, inverse of the forward difference operator ${\displaystyle \Delta }$. It relates to the forward difference operator as the indefinite integral relates to the derivative. Thus

${\displaystyle \Delta \sum _{x}f(x)=f(x)\,.}$

More explicitly, if ${\textstyle \sum _{x}f(x)=F(x)}$, then

${\displaystyle F(x+1)-F(x)=f(x)\,.}$

If F(x) is a solution of this functional equation for a given f(x), then so is F(x)+C(x) for any periodic function C(x) with period 1. Therefore, each indefinite sum actually represents a family of functions. However, due to the Carlson's theorem, the solution equal to its Newton series expansion is unique up to an additive constant C. This unique solution can be represented by formal power series form of the antidifference operator: ${\displaystyle \Delta ^{-1}={\frac {1}{e^{D}-1}}}$.