Indefinite sum

Indefinite sum

The inverse forward difference (blue) and inverse backward difference (yellow) of $x$.
Other names	Antidifference, inverse finite difference
Notation	$\sum _{x}f(x)$, $\Delta ^{-1}f(x)$ (forward), $\nabla ^{-1}f(x)$ (backward).
Definition	Linear operator that inverts the finite difference: $\Delta ^{-1}$: $\Delta F(x)=F(x+1)-F(x)=f(x)$ $\nabla ^{-1}$: $\nabla F(x)=F(x)-F(x-1)=f(x)$
Domain	Functions on integers; extended to real and complex arguments via analytic continuation
Uniqueness	Up to an arbitrary 1-periodic function; the Nørlund principal solution is unique up to a constant (minimal exponential type is imposed).
Video illustration	A 1‑periodic function $C(x)$ vanishes under the forward difference ($\Delta C(x)=0$).
Related concepts	Indefinite integral, Bohr–Mollerup theorem, finite difference, summation, Bernoulli polynomials, Gamma function, Euler–Maclaurin formula, Abel–Plana formula

In the calculus of finite differences, the indefinite sum (or antidifference operator), denoted by ${\textstyle \sum _{x}}$ or $\Delta ^{-1}$, is the linear operator that inverts the forward difference operator $\Delta f(x)=f(x+1)-f(x).$

That is, if ${\textstyle \sum _{x}f(x)=F(x)}$, then $F$ satisfies the functional equation $F(x+1)-F(x)$$=f(x)$ so that applying the forward difference recovers the original function: ${\textstyle \Delta \sum _{x}f(x)=f(x).}$ The operator thus plays the same role for finite differences that the indefinite integral plays for the derivative.

An indefinite sum is not unique: adding any 1-periodic function $C(x)$ (satisfying $C(x+1)=C(x)$), the function $F(x)+C(x)$ is also a solution. Therefore, an indefinite sum is unique up to a 1-periodic function $C(x)$ instead of up to a constant $C$ as the indefinite integral is.

To obtain the unique solution up to a constant $C$, one must impose additional analytic constraints. The Nørlund principal solution is the unique analytic solution that has the minimal possible exponential type (that is, its growth in the imaginary direction on the complex plane is the minimal possible), filtering out any non-constant periodic component. Other methods include higher-order convexity or concavity conditions in real analysis, or using axioms and complex analysis to step back the function's behavior from a neighborhood of infinity in which it behaves polynomially.

For integer arguments, the indefinite sum naturally extends ordinary summation, turning a discrete sum into a continuous function. Many such extensions are well-known special functions.