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Wikipedia / Indefinite sum

In the calculus of finite differences, the indefinite sum (or antidifference, is the linear operator that inverts the forward difference operator

so that applying the forward difference recovers the original function: The operator thus plays the same role for finite differences that the indefinite integral plays for the derivative.

as the indefinite integral is.

, 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.

Forward and backward difference conventions

:

:

. Stated explicitly, that is:

Which follows from the discrete counterpart:

Some authors use the equivalent form called the telescoping equation:

.

Fundamental theorem of the calculus of finite differences

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

Alternatively, using the inverse backward difference operator, the relation is:

Examples

represents an arbitrary 1-periodic function (or a constant if the Nørlund principal solution is assumed):

  1. Constant:

  2. Exponential:

  3. Logarithm:

  4. Powers (right half-plane):

are the generalized Bernoulli polynomials is the Hurwitz zeta function is the digamma function. This is related to the generalized harmonic numbers. Combined with series expansions (such as Taylor series expansions about a point or Laurent series expansions) or partial fraction decomposition, the power formula allows the indefinite summation of many analytic functions (term-wise, through the linearity of the operator).

In the calculus of finite differences, the power rule translates to the Bernoulli polynomials:

) when the constant is chosen to make the mean over a unit interval zero. In practice, falling factorials are more often used in contexts similar to the power rule.

Falling factorials

Falling factorials provide the discrete analog of the power rule from differential calculus.. In the calculus of finite differences, the falling factorial

, and the forward difference operator satisfies

The indefinite sum of a falling factorial is given by the discrete analog of the power rule for integration:

Equivalently, using the Gamma function:

, the solution is the digamma function, which naturally extends the harmonic numbers.

Example: and the indefinite sum formula above,

Applying the fundamental theorem of the calculus of finite differences,

Expanding the falling factorials,

and simplifying yields the formula

Summation by parts

Indefinite summation by parts is the discrete analog of integration by parts. It is derived from the product rule for the forward difference operator.

Product rule., the product rule for the forward difference is:

Introducing the shift operator, this can be written more compactly as:

Summation by parts. Rearranging the product rule gives:

is an arbitrary 1‑periodic function) yields the formula for summation by parts:

A symmetrical form, also obtained from the product rule, is:

Definite summation by parts., the formula becomes:

Example: product of a polynomial and an exponential

. Then:

Applying the summation by parts formula:

The remaining sum is elementary:

Hence the indefinite sum (antidifference) is

, we use the fundamental theorem with the forward difference inverse:

:

,

Uniqueness of the principal solution

.

To select the unique principal solution (German: Hauptlösung)) one must impose additional constraints.

Complex analysis (exponential type)

Following the theory developed by Niels Erik Nørlund, the indefinite sum can be uniquely determined for analytic functions by imposing restrictions on their growth in the complex plane. Specifically, by imposing minimal growth, the non-constant periodic terms can be filtered out. He states, ").

is analytic in a vertical strip containing a portion of the real line. has singularities (including those extending into the imaginary direction), a single vertical strip cannot contain the entire real axis. Instead, these singularities create vertical boundaries that split the domain into disjoint maximal vertical strips. into disjoint half-planes.

Nørlund’s theory provides a principal solution in each maximal vertical strip that contains a segment of the real line. While these infinite vertical strips can be shifted horizontally to evaluate the function, they cannot cross the singularities without the recurrence relation causing the singularities to repeat (e.g. digamma). Thus, each maximal vertical strip's principal solution contains no singularities in its respective maximal vertical strip, but contains singularities that recur outwards into outer disjoint maximal vertical strips.

The solution that contains the largest defined portion of the discrete sum being extended is typically designated as the "canonical" principal solution (treated as if it is the only principal solution). The choice of this strip and thus which principal solution is used is not uniquely dictated by the theory; rather, it is a matter of convention that varies between applications.

. Alternatively, authors may designate the strip situated immediately to the right of the rightmost singularity on the complex plane as the canonical strip, which allows the recurrence to propagate directly from the partial sums over positive integers to the right of said singularity.

is sufficient but not strictly necessary of slowest possible growth in the complex plane)..

in the denominator) the principal solution fails to exist (is undefined everywhere or fails to be unique up to a constant) because it resonates with the kernel of the difference operator:, on that maximal vertical strip.

For functions with isolated poles, distinct strips must give different branches: the native strip provides a pole-free principal solution, and the recurrence propagates the original poles by the step length of every neighboring strip, so no principal solution can be analytically continued across the vertical pole lines which correspond to a real part.

For functions with branch-point singularities (e.g. logarithms, positive fractional powers), the vertical strips do not impose a hard boundary on the solution. The indefinite sum may merge into a single-valued solution on a Riemann surface, where the potentially disjoint strips are actually connected by branch cuts. In such cases, the same principal solution can be valid on multiple sheets of the surface, and the number of independent principal solutions is equal to the number of distinct sheets (or branches) of the Riemann surface.

on each strip.

). Partial fractions give

never hit a non-positive integer, so the identity is valid at every term of the sum:

, the two terms are complex conjugates, and the expression simplifies to a real function:

-integer (or more generally, span size) shifts of the original singularities that lie in the left half-plane.

)., the analogous solution on the left half-plane is

, which lie in the right half-plane outside the original strip.

Summary: by integer steps, so any attempt to analytically continue one branch into the other strip introduces poles.

Real analysis (higher‑order convexity and concavity)

In real analysis, the uniqueness condition can be given using higher‑order convexity, generalizing the Bohr-Mollerup theorem-convex if its divided differences.

-concave.

Theorem.,

.

Müller–Schleicher axiomatic method

In their paper How to Add a Noninteger Number of Terms, Müller and Schleicher introduced an axiomatic approach to fractional summation with a real or complex number of terms. Their method extends the classical discrete sum

. The definition is built upon six natural axioms:

  1. .

  2. .

  3. .

  4. (equivalent to the empty sum condition).

  5. .

  6. , then:

    1. .

Axioms S1–S4 force the sum to align with the ordinary finite sum when the limits are integers. Axiom S5 forces monomials to behave the same way under the generalization of fractional sums. Axiom S6 is the crucial axiom that allows one to "step back" the asymptotic region to determine the fractional sum in a finite interval. The exact conditions for the method to work are, as stated in the Definition 1.2 of the paper:

will be called fractional summable of degree if the following conditions are satisfied:

  • the limit

exists.

(i.e., the approximating polynomials are zero), this reduces to:

Symmetry of the principal solution

is an entire function, one can define a unique analytic solution of the backward difference sum, by imposing the conditions that:

  • Difference Equation

  • Normalization (empty sum boundary condition).

  • Growth constraint has the minimal possible exponential type in the imaginary direction.

satisfies a reflection formula (referred to by Nørlund is the span).).

Odd functions

is an odd function exists, then

.

Even functions

is an even function, then

. Using the difference equation and oddness,

. . Then

. Consequently,

Choice of the constant term

must be fixed by an additional condition. Three common choices are the empty sum condition, an integral mean condition that identifies the result with the classical Bernoulli polynomials, and Ramanujan summation.

Empty sum boundary condition

The most direct method forces the indefinite sum to extend the usual discrete sum and to satisfy the empty sum convention.

  1. makes the sum over an empty interval zero.

    .

These conditions determine the solution uniquely up to an additive constant. For example,

.

Integral mean condition

In the study of Faulhaber's formula and the Euler–Maclaurin formula, it is convenient to identify the indefinite sum of a monomial with the corresponding Bernoulli polynomial.. The standard generating function

and summing term-wise via linearity), rather than serving as the defining property.

To match this convention, the constant is fixed by requiring that the solution have zero mean over a unit interval. For the inverse backward difference one may use

and for the inverse forward difference

Example., consistent with the Bernoulli normalization.

This normalization is not mandatory; in modern treatments the empty sum condition is usually preferred. This is usually used in the context of Bernoulli polynomials, the Hurwitz or Riemann zeta functions, the generalized harmonic number function, or other cases dealing with monomials and their antidifferences.

Relationship to indefinite products

In the symbolic method developed by Niels Erik Nørlund and L. M. Milne-Thomson, the indefinite product By taking the logarithm. This allows the indefinite product to be expressed through an indefinite sum:

is an arbitrary periodic function of period 1 assuming the disjoint maximal vertical strips and branch cuts of the natural logarithm is single-valued and its indefinite sum exists. Conversely, an indefinite sum may be represented as the logarithm of an indefinite product:

Gamma function and Gauss Pi function

alongside Nørlund's minimal growth. This yields the Gamma function

. Exponentiating and choosing the empty product condition to set the constant then yields the Gamma function.

, the solution is the Gauss Pi function.

recovers the classical factorial extensions.

Expansions and definitions

Abel–Plana formula

can be analytically continued by applying the standard Abel-Plana formula. This yields the formula:

This analytic continuation is valid when the conditions for the original formula are met. The sufficient conditions are:

after the scaling.

.. If such a strip cannot be established one must fall back to Laurent series or Taylor series expansions and sum term-wise. The exponential type can be reduced globally resonates in every possible strip.

Newton series

, can be expressed by its Newton series expansion:

  1. is the falling factorial.

Bernoulli‑operator series expansion

Formally using the exponential generating function of the Bernoulli numbers:

are the Bernoulli numbers.

one obtains the antidifference:

are the Bernoulli polynomials of the first order.

, the antidifference of monomials in the series expansion yields the formal series:

For non‑polynomials this expansion is generally asymptotic.

  1. Relation to the inverse backward difference

.

Euler–Maclaurin formula

, one has:

), and the remainder term is

gives the form

with the remainder

The formula gives the analytic continuation of the discrete sum.

Laplace summation (Gregory summation formula)

:

  1. are the Cauchy numbers of the first kind.

    is the falling factorial.

times a periodic Bernoulli polynomial. In the notation of Charles Jordan, Gregory's formula is:

are the Bernoulli numbers of the second kind. Note the argument is without a shift, aligning with the inverse backward difference.

Applications

The indefinite sum and its principal solution are fundamental to diverse fields, providing an analytic framework for extending discrete problems into the continuous and complex domains. Its applications span the construction of special functions, the regularization of divergent series in quantum field theory (e.g., the Casimir effect), performance analysis in queueing theory, scalar evolution in compilers such as LLVM or the GNU Compiler Collection, and various numerical and symbolic computational methods.

Special functions

Many standard transcendental functions are naturally defined as indefinite sums of elementary terms. Under the Nørlund principal solution—which imposes minimal exponential growth to eliminate arbitrary periodic components—these definitions coincide with the usual analytic continuations.

which generalizes the generalized harmonic numbers

Quantum field theory and the Casimir effect

In the mode-summation approach to the Casimir effect. The Abel–Plana formula—which provides an exact relation between a discrete sum and an integral—is a primary tool for isolating the divergent contribution. By subtracting the integral representation of the infinite-volume contribution, the formula yields a finite, cutoff-independent remainder for the Casimir energy and other observables. For geometries where the eigenfrequencies are zeros of Bessel functions, the ordinary Abel–Plana formula is insufficient. The generalized Abel–Plana formula extends the indefinite sum to summands with branch-point singularities and has been directly applied to regularise Casimir energies for spherical and cylindrical shells.

Queueing theory and probability

arise frequently in the analysis of stochastic queues. The Nørlund principal solution provides an explicit, analytically tractable solution for key performance measures, such as emptiness probabilities, waiting-time distributions, and blocking probabilities, while its minimal-growth condition automatically filters out non-physical, oscillatory 1-periodic components. servers, is naturally extended from integral to real trunk counts via the Nørlund sum of its defining recurrence.—which are essential for gradient-based sensitivity analysis and optimization of trunk-group sizing in telecommunications networks, as derivatives are otherwise undefined over discrete server counts. queue.

Numerical analysis and symbolic summation

The indefinite sum underpins several foundational computational methods:

  • The Euler–Maclaurin formula, used to accelerate series convergence and to estimate errors in numerical quadrature.

  • Gosper's algorithm and Karr's summation in finite terms find closed-form indefinite sums of hypergeometric terms and are implemented in computer algebra systems.

  • The Gregory-Laplace summation formula expresses the indefinite sum in terms of finite differences and is particularly useful for numerical integration over equispaced data, providing an efficient alternative to traditional quadrature when high-order differences decay rapidly.

Compiler scalar evolution

In compiler optimization, the scalar evolution (SCEV) analysis framework in LLVM and GCC utilizes chains of recurrences to model the values of induction variables, where the sum is a finite sum over the loop iteration count. These closed forms allow the compiler to replace repeated computations with direct formulas, enabling strength reduction, loop-invariant code motion, and dependence analysis without iterative re-evaluation of the recurrence.

See also

References

Further reading