In the calculus of finite differences, the indefinite sum, or antidifference, of a function is a solution of the functional equation For positive integer arguments, the indefinite sum reduces to ordinary summation. The function is thus the same kind of solution sought in fractional summation, which instead starts from the problem of extending finite sums to non-integer values of the summation bound.
Indefinite sums are generally not unique: if is an indefinite sum of then so is , where is a -periodic function.
By choosing one representative from the set of solutions in a way that depends linearly on , one obtains an indefinite summation or antidifference operator, often denoted with the greek letter Sigma , the summation symbol , or , which is a right inverse of the forward difference operator :Indefinite summation thus plays for finite differences a role analogous to indefinite integral for the derivative.
History The notation for indefinite summation goes back to Leonhard Euler, who in his 1755 Institutiones calculi differentialis introduced the symbol for the inverse operation to finite differencing . The notation was subsequently used by Joseph-Louis Lagrange in 1772. Indefinite summation is a predecessor of definite summation, introduced by Joseph Fourier in 1820.
Properties
Reduction to summation Indefinite sums can be used to calculate definite sums with the formula:
for integers and . As a convention, is often assumed in such a way that interpolates the partial sums .
If one wants to interpolate instead, then one should instead find a solution of the functional equation , satisfying , where is the backward difference operator.
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 for . For two functions and , the product rule for the forward difference is:
Summation by parts. Rearranging and taking the indefinite sum of both sides yields the formula for summation by parts:where is an arbitrary -periodic function.
Symmetry If is an odd function then which represents a point symmetry about . For example, gives .
If is an even function, then Examples
The following basic indefinite sums follow from the fundamental properties of the difference operator, where represents an arbitrary 1-periodic function:
Constant: Exponential: Integer powers:where is the th Bernoulli polynomial. Complex powers : where is the Hurwitz zeta function. Falling factorials:This can be further extended to complex using the Gamma function . Reciprocal function:where is the digamma function. This is related to the generalized harmonic numbers. Logarithm:
Defining a unique principal solution The functional equation does not have a unique solution. If is a particular solution, then for any function satisfying (i.e., any 1-periodic function), 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 (instead of up to the additive 1-periodic function ) one must impose additional constraints.
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
to non-integer and complex upper limits . The definition is built upon six natural axioms:
Continued Summation: . Translation Invariance: . Linearity: . Empty Sum Condition: (equivalent to the empty sum condition). Holomorphy for Monomials: for each , is holomorphic in . Right-Shift Continuity: if pointwise as , then ; more generally, if can be approximated by polynomials of fixed degree with , then: .
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:
Let and . A function will be called fractional summable of degree if the following conditions are satisfied: for all there exists a sequence of polynomials of fixed degree such that for all as for every the limit
exists.
In the simplest case when as (i.e., the approximating polynomials are zero), this reduces to:
Complex analytic method 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, "Unter allen regulären Lösungen sind also die Hauptlösungen die von kleinstem Wachstum in bezug auf " (Among all regular solutions, the principal solutions are therefore those of smallest growth with respect to ).
The usual formulation assumes that the summand is analytic in a vertical strip containing a portion of the real line. However, when 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. For example, poles at prevent a single strip from crossing the imaginary axis, splitting the domain at 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.
Two common practices exist. One approach selects the maximal vertical strip that contains the origin (or the strip immediately to its right if the origin lies on a singularity), as this often preserves symmetry or aligns with the natural boundary condition . 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.
Pole placement
For a given maximal vertical strip , the singularities of the principal solution outside the strip are determined by the singularities of and the direction of the recursion. Let denote the step size, and let be a singular point of . The placement of the poles of is governed by the following rules.
For the inverse backward difference , Nørlund's analysis shows that the periodic function is subtracted from the bare solution to obtain the principal solution. This subtraction regularizes the term on the left boundary while retaining it on the right boundary.: "Es wird also ."
Domain Poles of (left boundary) , (right boundary) ,
For the inverse forward difference , Nørlund's analysis shows that the periodic function is subtracted from the bare solution. This subtraction regularizes the term on the right boundary while retaining it on the left boundary.: "Es wird also ."
Domain Poles of (left boundary) , (right boundary) ,
The asymmetry arises because the recursion for the backward difference steps left, "landing" on the right boundary and retaining the pole there, while the forward difference steps right, "landing" on the left boundary and retaining the pole there. In both cases, the interior of the strip is completely clean (no poles), all singularities of outside the strip are shifted outward by integer multiples of in the appropriate direction, and the solution picks up all singularities outside the strip, not just those on the immediate boundaries.
The condition of exponential type less than in the imaginary direction on is sufficient but not strictly necessary. Nørlund's general definition of the principal solution is the analytic solution having Fourier components of the minimal possible exponential type for the given ( of slowest possible growth in the complex plane). If has exponential type in imaginary direction, then the principal solution will also have type in that strip, provided it converges. For example, has exponential type ; its principal solution exists and has type , even though .
When has exponential type exactly for some non-zero integer in every strip where it is analytic (e.g. has type ; its antidifference contains 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:
In all other cases–when is meromorphic and on some vertical strip that contains a segment of the real line, and its exponential type on said strip is not an integer multiple of –the principal solution exists and is uniquely determined, up to a constant , on that maximal vertical strip.
For functions with isolated poles, distinct strips must give different families of solutions: the native strip provides a pole-free principal solution, and the recurrence propagates the original poles by the step length of every neighboring strip. Thus, the principal solution for a given strip is not the principal solution for an adjacent strip, even if it can be analytically continued across the boundary. The pole placement rules determine the unique minimal-type solution for each open interval. When branch points or logarithms are involved, the boundaries can be more subtle, and the number of unique principal solutions may relate to the distinct sheets of the resulting Riemann surface, rather than simply the count of isolated poles.
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.
Consider the meromorphic function . Its poles at split the complex plane into two maximal vertical strips that each contain a segment of the real line: the right half-plane and the left half-plane . We construct the Nørlund principal solution of the backward difference equation with empty-sum normalization on each strip.
Right half-plane (). Partial fractions give
The digamma function satisfies for all . For the arguments and never hit a non-positive integer, so the identity is valid at every term of the sum:
Because , the two terms are complex conjugates, and the expression simplifies to a real function: