Indefinite sum
In the calculus of finite differences, the indefinite sum (or antidifference operator), denoted by ∑ x {\textstyle \sum _{x}} or Δ − 1 {\displaystyle \Delta ^{-1}} , is the linear operator that inverts the forward difference operator Δ f ( x ) = f ( x + 1 ) − f ( x ) . {\displaystyle \Delta f(x)=f(x+1)-f(x).} That is, if ∑ x f ( x ) = F ( x ) {\displaystyle \sum _{x}f(x)=F(x)} , then F {\displaystyle F} satisfies the functional equation F ( x + 1 ) − F ( x ) {\displaystyle F(x+1)-F(x)} = f ( x ) , {\displaystyle =f(x),} so that applying the forward difference recovers the original function: Δ ∑ x f ( x ) = f ( x ) .