Indefinite sum

In the calculus of finite differences, the indefinite sum operator, denoted by or, is the linear operator that is the inverse of the forward difference operator. It relates to the forward difference operator as the indefinite integral relates to the derivative. Thus,
More explicitly, if, then
The solution is not unique; it is determined only up to an additive periodic function with period 1. Therefore, each indefinite sum represents a family of functions.

Fundamental theorem of the calculus of finite differences

Indefinite sums can be used to calculate definite sums with the formula:

Alternative usage

The inverse forward difference operator,, extends the summation up to :
Some authors analytically extend summation for which the upper limit is the argument without a shift:
In this case, a closed-form expression for the sum is a solution of
which is called the telescoping equation. It is the inverse of the backward difference operator, :
It is related to the forward antidifference operator using the fundamental theorem of the calculus of finite differences.

Uniqueness

The functional equation does not have a unique solution. If is a particular solution, then for any function satisfying , the function is also a solution. Therefore, the indefinite sum operator defines a family of functions differing by an arbitrary 1-periodic component,.
To select the unique principal solution up to an additive constant one must impose additional constraints.

Complex analysis (Exponential type)

Suppose is analytic in a vertical strip containing the real axis, and let be an analytic solution of in that strip. To ensure uniqueness, we require to be of minimal growth, specifically to be of exponential type less than in the imaginary direction. That is, there exist constants and such that
as.
Now let and be two analytic solutions satisfying this growth condition. Their difference is then analytic, 1‑periodic, and inherits the same exponential type less than.
A fundamental result in complex analysis states that a non‑constant 1‑periodic entire function must have exponential type at least. This follows from its Fourier series expansion: if is non‑constant, its Fourier series contains a term with, which has type. Since has type strictly less than, it cannot contain any such term and therefore must be constant.
Consequently, under this minimal growth condition, any two solutions differ by at most a constant. Hence is unique up to an additive constant.

Relationship to Indefinite products

The indefinite product operator, denoted by, is the multiplicative analogue of the indefinite sum. If, then:
Its common discrete analog is. The two operators are related by:

Expansions and Definitions

Laplace summation formula

The Laplace summation formula is a formal asymptotic expansion of the inverse forward difference :

Newton series

The inverse forward difference operator,, can be expressed formally by its Newton series expansion:

Faulhaber's formula

Given that can be represented by its Maclaurin Series expansion, the Taylor series about, it is sometimes possible to represent the indefinite sum using Bernoulli polynomials because :

Müller-Schleicher Axiomatic definition

If is analytic on the right half-plane and satisfies the decay condition, the analytic continuation of is given by:
This formula is derived from axioms presented in the paper based on fractional sums, which uniquely extends the definition of the summation to complex limits. The decay condition represents the simplest case of the general asymptotic requirements for the function.

Euler–Maclaurin formula

The Euler–Maclaurin formula extends :
where are the even Bernoulli numbers, is an arbitrary positive integer, and is the remainder term given by:
with being the periodized Bernoulli function related to the Bernoulli polynomials.

Abel–Plana formula

The indefinite sum can be analytically continued by applying the standard Abel-Plana formula to the finite sum and then analytically continuing the integer limit to the variable. This yields the formula:
This analytic continuation is valid when the conditions for the original formula are met. The sufficient conditions are:

Analyticity: must be analytic in the closed vertical strip between and. The formula provides analytic continuation up to, but not beyond, the nearest singularities of to the line.
Growth: must be of exponential type less than in this strip, satisfying for some, as.
Choice of the constant term

Analytic Continuation of Discrete Sums

The constant term, in the context of indefinite sums naturally extending the discrete summation, is often defined based on the respective empty sum.
For the inverse forward difference,, the typical summation equivalent is so the empty sum is when as it correlates to
For the inverse backward difference,, the typical summation equivalent is so the empty sum is when as it correlates to

Normalization

In older texts relating to Bernoulli polynomials the constant was often fixed using integral conditions.
Let
Then the constant is fixed from the condition
or
Alternatively, Ramanujan summation can be used:
or at 1
respectively.

Summation by parts

Indefinite summation by parts:
Definite summation by parts:

Period rules

If is a period of function then
If is an antiperiod of function, that is then

Reflection Formulas (Parity Rules)

The unique analytic continuation of defined as
with exponential type less than in the imaginary direction where is entire and the constant term is chosen such that , satisfies a reflection formula.

Odd Functions

If is an odd function, the unique analytic continuation satisfies:
This represents a point symmetry about.

Even Functions

If is an even function, the unique analytic continuation satisfies:

List of indefinite sums

Antidifferences of rational functions

For positive integer exponents, Faulhaber's formula can be used. Note that in the result of Faulhaber's formula must be replaced with due to the offset, as Faulhaber's formula finds rather than.
For negative integer exponents, the indefinite sum is closely related to the polygamma function:
For fractions not listed in this section, one may use the polygamma function with partial fraction decomposition. More generally,
where are the Bernoulli polynomials, is the Hurwitz zeta function, and is the digamma function. This is related to the generalized harmonic numbers.
As the generalized harmonic numbers use reciprocal powers, must be substituted for, and the most common form uses the inverse of the backward difference offset:
Here, is the constant.
The Bernoulli polynomials are also related via a partial derivative with respect to :
Similarly, using the inverse of the backwards difference operator may be considered more natural, as:
Further generalization comes from use of the Lerch transcendent:
which generalizes the generalized harmonic numbers as when taking. Additionally, the partial derivative is given by