Indefinite sum

Reduction to summation

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

${\displaystyle \sum _{k=a}^{b-1}f(k)=\Delta ^{-1}f(b)-\Delta ^{-1}f(a),}$

for integers ${\displaystyle a}$ and ${\displaystyle b}$. As a convention, ${\displaystyle \Delta ^{-1}f(0)=0}$ is often assumed in such a way that ${\displaystyle \Delta ^{-1}f(x)}$ interpolates the partial sums ${\textstyle (\sum _{k=0}^{n-1}f(k))_{n\in \mathbb {N} }}$.

If one wants to interpolate ${\textstyle (\sum _{k=1}^{n}f(k))_{n\in \mathbb {N} }}$ instead, then one should instead find a solution ${\displaystyle \nabla ^{-1}f(x)}$ of the functional equation ${\textstyle \nabla F(x)=f(x)}$, satisfying ${\displaystyle \nabla ^{-1}f(0)=0}$, where ${\textstyle \nabla }$ 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 ${\textstyle \Delta }$. For two functions ${\displaystyle u(x)}$ and ${\displaystyle v(x)}$, the product rule for the forward difference is:

${\displaystyle \Delta (u(x)v(x))=u(x)\Delta v(x)+v(x+1)\Delta u(x).}$

Summation by parts. Rearranging and taking the indefinite sum of both sides yields the formula for summation by parts:$${\displaystyle \sum u(x)\Delta v(x)=u(x)v(x)-\sum v(x+1)\Delta u(x)+C(x),}$$where ${\displaystyle C(x)}$ is an arbitrary ${\displaystyle 1}$-periodic function.

Symmetry

If ${\displaystyle f}$ is an odd function then ${\textstyle F(z)=F(-1-z),}$ which represents a point symmetry about ${\displaystyle x=-1/2}$. For example, ${\displaystyle f(z)=z}$ gives ${\displaystyle F(z)=z(z+1)/2}$.

If ${\displaystyle f}$ is an even function, then ${\textstyle F(z)=F(-1)-F(-1-z).}$

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

${\displaystyle \sum _{k=1}^{x}f(k)}$

to non-integer and complex upper limits ${\displaystyle x}$. The definition is built upon six natural axioms:

1. Continued Summation: ${\displaystyle \sum _{\nu =x}^{y}f(\nu )+\sum _{\nu =y+1}^{z}f(\nu )=\sum _{\nu =x}^{z}f(\nu )}$.
2. Translation Invariance: ${\displaystyle \sum _{\nu =x+s}^{y+s}f(\nu )=\sum _{\nu =x}^{y}f(\nu +s)}$.
3. Linearity: ${\displaystyle \sum _{\nu =x}^{y}(\lambda f(\nu )+\mu g(\nu ))=\lambda \sum _{\nu =x}^{y}f(\nu )+\mu \sum _{\nu =x}^{y}g(\nu )}$.
4. Empty Sum Condition: ${\displaystyle \sum _{\nu =1}^{1}f(\nu )=f(1)}$ (equivalent to the empty sum condition).
5. Holomorphy for Monomials: for each ${\displaystyle d\in \mathbb {N} }$, ${\displaystyle z\mapsto \sum _{\nu =1}^{z}\nu ^{d}}$ is holomorphic in ${\displaystyle \mathbb {C} }$.
6. Right-Shift Continuity: if ${\displaystyle f(z+n)\to 0}$ pointwise as ${\displaystyle n\to +\infty }$, then ${\displaystyle \sum _{\nu =x}^{y}f(\nu +n)\to 0}$; more generally, if ${\displaystyle f(z+n)}$ can be approximated by polynomials ${\displaystyle p_{n}(z+n)}$ of fixed degree with ${\displaystyle |f(z+n)-p_{n}(z+n)|\to 0}$, then:

${\displaystyle \left|\sum _{\nu =x}^{y}f(\nu +n)-\sum _{\nu =x}^{y}p_{n}(\nu +n)\right|\to 0}$.

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 ${\displaystyle U\subset \mathbb {C} }$ and ${\displaystyle \sigma \in \mathbb {N} \cup \{-\infty \}}$. A function ${\displaystyle f:U\rightarrow \mathbb {C} }$ will be called fractional summable of degree ${\displaystyle \sigma }$ if the following conditions are satisfied:

• ${\displaystyle x+1\in U}$ for all ${\displaystyle x\in U;}$
• there exists a sequence of polynomials ${\displaystyle (p_{n})_{n\in \mathbb {N} }}$ of fixed degree ${\displaystyle \sigma }$ such that for all ${\displaystyle x\in U}$

${\displaystyle |f(n+x)-p_{n}(n+x)|\rightarrow 0}$ as ${\displaystyle n\rightarrow +\infty }$

• for every ${\displaystyle x,y+1\in U,}$ the limit

${\displaystyle \lim _{n\to \infty }\left(\sum _{\nu =n+x}^{n+y}p_{n}(\nu )+\sum _{\nu =1}^{n}{\bigl (}f(\nu +x-1)-f(\nu +y){\bigr )}\right),}$

exists.

In the simplest case when ${\displaystyle f(t)\to 0}$ as ${\displaystyle t\to \infty }$ (i.e., the approximating polynomials are zero), this reduces to:

${\displaystyle \nabla ^{-1}f(x)=\sum _{k=1}^{x}f(k)=\sum _{n=1}^{\infty }{\bigl (}f(n)-f(n+x){\bigr )}+C}$

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 ${\displaystyle x}$" (Among all regular solutions, the principal solutions are therefore those of smallest growth with respect to ${\displaystyle x}$).

The usual formulation assumes that the summand ${\displaystyle f(z)}$ is analytic in a vertical strip containing a portion of the real line. However, when ${\displaystyle f(z)}$ 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 ${\displaystyle \pm i}$ prevent a single strip from crossing the imaginary axis, splitting the domain at ${\displaystyle \Re (z)=0}$ 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 ${\displaystyle F(0)=0}$. 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 ${\displaystyle (a,b)}$, the singularities of the principal solution ${\displaystyle F}$ outside the strip are determined by the singularities of ${\displaystyle f}$ and the direction of the recursion. Let ${\displaystyle S}$ denote the step size, and let ${\displaystyle \beta }$ be a singular point of ${\displaystyle f}$. The placement of the poles of ${\displaystyle F}$ is governed by the following rules.

For the inverse backward difference ${\displaystyle F(z)-F(z-S)=f(z)}$, Nørlund's analysis shows that the periodic function ${\displaystyle {\mathfrak {R}}(x|\omega )}$ is subtracted from the bare solution to obtain the principal solution. This subtraction regularizes the ${\displaystyle m=0}$ term on the left boundary while retaining it on the right boundary.

Domain	Poles of ${\displaystyle F}$
${\displaystyle \Re (\beta )\leq a}$ (left boundary)	${\displaystyle \beta -mS}$, ${\displaystyle m=1,2,3,\ldots }$
${\displaystyle \Re (\beta )\geq b}$ (right boundary)	${\displaystyle \beta +mS}$, ${\displaystyle m=0,1,2,\ldots }$

For the inverse forward difference ${\displaystyle F(z+S)-F(z)=f(z)}$, Nørlund's analysis shows that the periodic function ${\displaystyle \Pi (x|\omega )}$ is subtracted from the bare solution. This subtraction regularizes the ${\displaystyle m=0}$ term on the right boundary while retaining it on the left boundary.

Domain	Poles of ${\displaystyle F}$
${\displaystyle \Re (\beta )\leq a}$ (left boundary)	${\displaystyle \beta -mS}$, ${\displaystyle m=0,1,2,\ldots }$
${\displaystyle \Re (\beta )\geq b}$ (right boundary)	${\displaystyle \beta +mS}$, ${\displaystyle m=1,2,3,\ldots }$

The asymmetry arises because the recursion for the backward difference steps left, "landing" on the right boundary and retaining the ${\displaystyle m=0}$ pole there, while the forward difference steps right, "landing" on the left boundary and retaining the ${\displaystyle m=0}$ pole there. In both cases, the interior of the strip is completely clean (no poles), all singularities of ${\displaystyle f}$ outside the strip are shifted outward by integer multiples of ${\displaystyle S}$ 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 ${\displaystyle 2\pi }$ in the imaginary direction on ${\displaystyle f}$ is sufficient but not strictly necessary. Nørlund's general definition of the principal solution is the analytic solution ${\displaystyle F}$ having Fourier components of the minimal possible exponential type for the given ${\displaystyle f}$ (${\displaystyle F}$ of slowest possible growth in the complex plane). If ${\displaystyle f}$ has exponential type ${\displaystyle k}$ in imaginary direction, then the principal solution ${\displaystyle F(z)}$ will also have type ${\displaystyle k}$ in that strip, provided it converges. For example, ${\displaystyle f(z)=\sin(7z)}$ has exponential type ${\displaystyle 7}$; its principal solution exists and has type ${\displaystyle 7}$, even though ${\displaystyle 7>2\pi }$.

When ${\displaystyle f}$ has exponential type exactly ${\displaystyle 2\pi n}$ for some non-zero integer ${\displaystyle n}$ in every strip where it is analytic (e.g. ${\displaystyle f(z)=\sin(2\pi nz)}$ has type ${\displaystyle 2\pi n}$; its antidifference contains ${\displaystyle \sin(\pi n)=0}$ 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: $${\displaystyle \Delta ^{-1}={\frac {1}{e^{D}-1}}.}$$ In all other cases–when ${\displaystyle f}$ is meromorphic and on some vertical strip that contains a segment of the real line, and its exponential type on said strip is not a nonzero integer multiple of ${\displaystyle 2\pi }$–the principal solution exists and is uniquely determined, up to a constant ${\displaystyle C}$, 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.

Example: Partitioning disjoint maximal vertical strips (${\displaystyle \nabla ^{-1}1/(x^{2}+1)}$)

Consider the meromorphic function ${\displaystyle f(x)=1/(x^{2}+1)}$. Its poles at ${\displaystyle x=\pm i}$ split the complex plane into two maximal vertical strips that each contain a segment of the real line: the right half-plane ${\displaystyle \Re (x)>0}$ and the left half-plane ${\displaystyle \Re (x)<0}$. We construct the Nørlund principal solution of the backward difference equation ${\displaystyle F(x)-F(x-1)=f(x)}$ with empty-sum normalization ${\displaystyle F(0)=0}$ on each strip. Right half-plane (${\displaystyle \Re (x)>0}$). Partial fractions give ${\displaystyle {\frac {1}{x^{2}+1}}={\frac {1}{2i}}\!\left({\frac {1}{x-i}}-{\frac {1}{x+i}}\right).}$ The digamma function satisfies ${\displaystyle \psi (z+1)-\psi (z)=1/z}$ for all ${\displaystyle z\notin \{0,-1,-2,\dots \}}$. For ${\displaystyle \Re (x)>0}$ the arguments ${\displaystyle x+1\pm i}$ and ${\displaystyle 1\pm i}$ never hit a non-positive integer, so the identity is valid at every term of the sum: ${\displaystyle \sum _{k=1}^{x}{\frac {1}{k^{2}+1}}={\frac {1}{2i}}{\Bigl [}{\bigl (}\psi (x+1-i)-\psi (1-i){\bigr )}-{\bigl (}\psi (x+1+i)-\psi (1+i){\bigr )}{\Bigr ]}.}$ Because ${\displaystyle \psi ({\overline {z}})={\overline {\psi (z)}}}$, the two terms are complex conjugates, and the expression simplifies to a real function: ${\displaystyle F_{\text{right}}(x)=\operatorname {Im} \psi (1+i)-\operatorname {Im} \psi (x+1+i),\qquad \Re (x)>0.}$ The non-simplified function is analytic on the whole right half-plane, and ${\displaystyle F_{\text{right}}(0)=0}$. Within this strip the difference equation ${\displaystyle F(x)-F(x-1)=1/(x^{2}+1)}$ holds for every ${\displaystyle x}$ in the respective maximal vertical strip (half-plane). If one attempts to continue ${\displaystyle F_{\text{right}}}$ across the imaginary axis by the recurrence ${\displaystyle F(x+1)=F(x)+f(x+1)}$, poles appear at ${\displaystyle x=\pm i-1,\pm i-2,\dots }$-integer (or more generally, span size) shifts of the original singularities that lie in the left half-plane. Left half-plane (${\displaystyle \Re (x)<0}$). Using the reflection ${\displaystyle x\mapsto -x}$, the analogous solution on the left half-plane is ${\displaystyle F_{\text{left}}(x)=\operatorname {Im} \psi (-x+i)-\operatorname {Im} \psi (i),\qquad \Re (x)<0.}$ For ${\displaystyle \Re (x)<0}$ the argument ${\displaystyle -x+i}$ has a positive real part, so it never equals ${\displaystyle 0,-1,-2,\dots }$; the digamma identity applies and the difference equation is satisfied for all ${\displaystyle x}$ in the strip. Again the non-simplified function is analytic on the whole left half-plane. Extending this solution to the right via the recurrence would introduce poles at ${\displaystyle x=\pm i,\pm i+1,\pm i+2,\dots }$, which lie in the right half-plane outside the original strip. Summary: Domain Principal solution (inverse backward difference, ${\displaystyle F(0)=0}$) ${\displaystyle \Re (x)>0}$ ${\displaystyle \operatorname {Im} \psi (1+i)-\operatorname {Im} \psi (x+1+i)}$ ${\displaystyle \Re (x)<0}$ ${\displaystyle \operatorname {Im} \psi (-x+i)-\operatorname {Im} \psi (i)}$ The two expressions are analytic on their respective strips and give distinct principal solutions. The poles of the digamma function, which would violate the identity ${\displaystyle \psi (z+1)-\psi (z)=1/z}$, are never reached inside the respective domains. However, the recurrence propagates the original singularities of ${\displaystyle f}$ by integer steps, so any attempt to analytically continue one branch into the other strip introduces poles.

Convex analytic method

In real analysis, the uniqueness condition can be given using higher‑order convexity, generalizing the Bohr-Mollerup theorem. For an integer ${\displaystyle p\geq 0}$, a function is called ${\displaystyle p}$-convex if its divided differences of order ${\displaystyle p+1}$ (i.e., divided differences computed from ${\displaystyle p+2}$ distinct points) are non‑negative, and ${\displaystyle p}$-concave if those divided differences are non-positive. A function is called eventually ${\displaystyle p}$-convex (resp. eventually ${\displaystyle p}$-concave) if there exists ${\displaystyle M>0}$ such that it is ${\displaystyle p}$-convex (resp. ${\displaystyle p}$-concave) on the interval ${\displaystyle (M,\infty )}$.

Marichal and Zenaïdi proved the following uniqueness theorem, their method requiring the solution to be eventually ${\displaystyle p}$-convex or ${\displaystyle p}$-concave.

Theorem. Let ${\displaystyle p\geq 0}$ be an integer and let ${\displaystyle g:\mathbb {R} _{+}\to \mathbb {R} }$ satisfy ${\displaystyle \lim _{n\to \infty }\Delta ^{p}g(n)=0}$. If ${\displaystyle f:\mathbb {R} _{+}\to \mathbb {R} }$ is an eventually ${\displaystyle p}$-convex or eventually ${\displaystyle p}$-concave solution of ${\displaystyle \Delta f=g}$, then ${\displaystyle f}$ is uniquely determined up to an additive constant. Moreover, for any ${\displaystyle x>0}$,

${\displaystyle f(x)=f(1)+\lim _{n\to \infty }\left(\sum _{k=1}^{n-1}g(k)-\sum _{k=0}^{n-1}g(x+k)+\sum _{j=1}^{p}{\binom {x}{j}}\Delta ^{\,j-1}g(n)\right),}$

and the convergence is uniform on bounded subsets of ${\displaystyle \mathbb {R} _{+}}$.

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. Alternatively, ${\displaystyle \lim _{x\to 0^{-}}F(x)=0\ {\text{or}}\ \lim _{x\to 0^{+}}F(x)=0.}$

${\displaystyle \Delta ^{-1}f(x)}$ corresponds to ${\displaystyle \textstyle \sum _{k=0}^{x-1}f(k)}$. The convention ${\displaystyle \left.\Delta ^{-1}f(x)\right|_{x=0}=0}$ makes the sum over an empty interval zero.

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

${\displaystyle \nabla ^{-1}x^{a}={H_{x}^{(-a)}}=\zeta (-a)-\zeta (-a,x+1).}$

Here, ${\displaystyle \zeta (-a)}$ is the constant ${\displaystyle C}$ such that ${\displaystyle F(0)=0}$.

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. Following Nørlund's construction, the Bernoulli polynomials are obtained as the unique polynomial solution to the discrete difference equation ${\displaystyle \Delta B_{\nu }(x)=\nu x^{\nu -1}}$, subject to the initial condition ${\displaystyle B_{\nu }(0)=B_{\nu }}$. From this definition one derives the analytic differentiation rule ${\displaystyle {\frac {d}{dx}}B_{\nu }(x)=\nu B_{\nu -1}(x)}$, which directly yields the integration formula ${\displaystyle \int _{x}^{x+1}B_{\nu }(z)\,dz=x^{\nu }}$ and, consequently, the normalization condition ${\displaystyle \int _{0}^{1}B_{n}(x)\,dx=0}$. The standard generating function

${\displaystyle {\frac {te^{xt}}{e^{t}-1}}=\sum _{n=0}^{\infty }B_{n}(x){\frac {t^{n}}{n!}}}$

is then derived as a formal consequence of these prior results (specifically, by applying the formal operator identity ${\displaystyle \Delta ^{-1}=(e^{D}-1)^{-1}}$ to the Maclaurin series expansion of ${\displaystyle e^{xt}}$ 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

${\displaystyle \int _{-1}^{0}{\bigl (}\nabla ^{-1}f(x)+C{\bigr )}\,dx=0\quad {\text{or}}\quad \int _{0}^{1}{\bigl (}\nabla ^{-1}f(x)+C{\bigr )}\,dx=0,}$

and for the inverse forward difference

${\displaystyle \int _{0}^{1}{\bigl (}\Delta ^{-1}f(x)+C{\bigr )}\,dx=0\quad {\text{or}}\quad \int _{1}^{2}{\bigl (}\Delta ^{-1}f(x)+C{\bigr )}\,dx=0.}$

Example. For ${\displaystyle \Delta ^{-1}x={\frac {x(x-1)}{2}}+C}$, the condition ${\displaystyle \int _{0}^{1}(\Delta ^{-1}x)\,dx=0}$ gives ${\displaystyle C={\tfrac {1}{12}}}$. Hence ${\displaystyle \Delta ^{-1}x={\frac {x(x-1)}{2}}+{\frac {1}{12}}={\tfrac {1}{2}}B_{2}(x)}$ with ${\displaystyle B_{2}(x)=x^{2}-x+{\tfrac {1}{6}}}$, 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.

Gamma function and Gauss Pi function

For ${\displaystyle f(x)=x}$, the forward difference indefinite product ${\displaystyle \prod x}$ is the solution of ${\displaystyle F(x+1)=x\,F(x)}$ with the empty product normalization ${\displaystyle F(1)=1}$ alongside Nørlund's minimal growth. This yields the Gamma function ${\displaystyle \prod x=\Gamma (x),}$ which represents the discrete product ${\displaystyle \prod _{k=1}^{x-1}k.}$

It can also be obtained by taking the Nørlund principal solution to the inverse forward difference: ${\displaystyle \sum \log x=\log \Gamma (x)+C(x)}$. Exponentiating and choosing the empty product condition to set the constant then yields the Gamma function.

If one instead works with the backward difference indefinite product, satisfying ${\displaystyle F(x)=x\,F(x-1)}$ and ${\displaystyle F(0)=1}$, the solution is the Gauss Pi function, ${\displaystyle \Pi (x)}$, which extends ${\displaystyle x!}$ directly and represents the discrete product ${\displaystyle \prod _{k=1}^{x}k}$.

The two conventions differ only by a shift of the argument (a property inherited from the relationship between ${\displaystyle \nabla ^{-1}}$ and ${\displaystyle \Delta ^{-1}}$): ${\displaystyle \Pi (x)=\Gamma (x+1)}$. The minimal imaginary growth condition in each case forces ${\displaystyle \varpi (x)}$ to be a constant ${\displaystyle 1}$, so the principal indefinite product of ${\displaystyle x}$ recovers the classical factorial extensions.

Abel–Plana formula

The indefinite sum ${\displaystyle \nabla ^{-1}f(x)=\sum _{k=1}^{x}f(k)}$ can be analytically continued by applying the standard Abel-Plana formula to the finite sum ${\displaystyle \sum _{k=1}^{n}f(k)}$ and then analytically continuing the integer limit ${\displaystyle n}$ to the variable ${\displaystyle x}$. This yields the formula: $${\displaystyle {\begin{aligned}\nabla ^{-1}f(x)&=\int _{1}^{x}f(t)dt+{\frac {f(1)+f(x)}{2}}\\&\quad +i\int _{0}^{\infty }{\frac {\left(f(x-it)-f(1-it)\right)-\left(f(x+it)-f(1+it)\right)}{e^{2\pi t}-1}}dt+C(x)\end{aligned}}}$$This analytic continuation is valid when the conditions for the original formula are met. The sufficient conditions are:

1. Analyticity: ${\displaystyle f(z)}$ must be analytic in the closed vertical strip between ${\displaystyle \Re (z)=1}$ and ${\displaystyle \Re (z)=\Re (x)}$. The formula provides the analytic solution up to, but not beyond, the nearest singularities of ${\displaystyle f}$ to the line ${\displaystyle \Re (z)=1.}$
2. Growth: ${\displaystyle f(z)}$ must be of exponential type less than ${\displaystyle 2\pi }$ in this strip, satisfying ${\displaystyle |f(z)|\leq Me^{(2\pi -\epsilon )|\Im (z)|}}$ for some ${\displaystyle M>0}$, ${\displaystyle \epsilon >0}$ as ${\displaystyle |\Im (z)|\to \infty .}$

The Abel–Plana formula can be used to obtain the Nørlund principal solution on arbitrary disjoint maximal vertical strips through the recurrence ${\displaystyle F(z)-F(z-1)=f(z)}$ allowing one to shift the Abel–Plana formula's domain of convergence, then repeatedly extend out of the original domain of convergence using the known valid domain of ${\displaystyle F}$ (which is known on the starting maximal vertical strip) and ${\displaystyle f}$ (which is entirely known), again justified through rearrangement of the equation ${\displaystyle F(z)-F(z-1)=f(z)}$. This is under the condition that a step size length (along the real axis) maximal vertical strip (extending infinitely in the imaginary directions) can be obtained while being singularity free and of less than ${\displaystyle 2\pi /S}$ (where ${\displaystyle S}$ is the step size) exponential type in the imaginary direction on ${\displaystyle f}$. 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 through summation by parts rules and the linearity of the operator to make ${\displaystyle f}$ possible to take the antidifference numerically using Abel–Plana and term-wise expansions, unless the ${\displaystyle f}$ globally resonates in every possible strip.

Newton series

For an entire function of exponential type less than ${\displaystyle \log \left(2\right)}$ the inverse forward difference operator, ${\displaystyle \Delta ^{-1}f(x)}$, can be expressed by its Newton series expansion:

${\displaystyle \sum f(x)=\sum _{k=1}^{\infty }{\binom {x}{k}}\Delta ^{k-1}f\left(0\right)+C(x)=\sum _{k=1}^{\infty }{\frac {\Delta ^{k-1}f(0)}{k!}}(x)_{k}+C(x).}$

${\displaystyle (x)_{k}={\frac {\Gamma (x+1)}{\Gamma (x-k+1)}}}$ is the falling factorial.

Bernoulli‑operator series expansion

Formally, the inverse forward difference operator can be expressed in terms of the derivative operator ${\displaystyle D={\frac {d}{dx}}}$ using the exponential generating function of the Bernoulli numbers:$${\displaystyle \Delta ^{-1}={\frac {1}{e^{D}-1}}=\sum _{v=0}^{\infty }{\frac {B_{v}}{v!}}D^{\,v-1},}$$where ${\displaystyle B_{v}}$ are the Bernoulli numbers defined by the generating function ${\displaystyle {\frac {t}{e^{t}-1}}=\sum _{v=0}^{\infty }B_{v}{\frac {t^{v}}{v!}}}$. Under this convention ${\displaystyle B_{1}=-{\tfrac {1}{2}}}$.

If ${\displaystyle f}$ is a polynomial, only finitely many terms of the series are non-zero as the finite difference of a monomial is a polynomial of one degree lower (following by induction, finitely many terms are required). For ${\displaystyle f(x)=x^{n}}$ one obtains the antidifference:$${\displaystyle \sum x^{n}={\frac {B_{n+1}(x)}{n+1}}+C(x),}$$where ${\displaystyle B_{n}(x)}$ are the Bernoulli polynomials of the first order.

If ${\displaystyle f}$ admits a Maclaurin series expansion ${\displaystyle f(x)=\sum _{n=0}^{\infty }{\frac {f^{(n)}(0)}{n!}}x^{n}}$, the antidifference of monomials in the series expansion yields the formal series:$${\displaystyle \sum f(x)=\sum _{n=1}^{\infty }{\frac {f^{(n-1)}(0)}{n!}}B_{n}(x)+C(x).}$$For non‑polynomials this expansion is generally asymptotic.

Relation to the inverse backward difference

If one instead expands the inverse backward difference operator, ${\displaystyle \nabla ^{-1}={\frac {e^{D}}{e^{D}-1}}}$ (which extends ${\displaystyle \sum _{k=1}^{x}f(k)}$), it admits the same expansion, but with ${\displaystyle B_{1}=+{\tfrac {1}{2}}}$ in place of ${\displaystyle B_{1}=-{\tfrac {1}{2}}}$.

Euler–Maclaurin formula

The Euler–Maclaurin formula provides an asymptotic expansion for the inverse backward difference ${\displaystyle \nabla ^{-1}f(x)=\sum _{k=1}^{x}f(k)}$ when the function is sufficiently smooth. For any positive integer ${\displaystyle m}$, one has:$${\displaystyle {\begin{aligned}\nabla ^{-1}f(x)&=\int _{1}^{x}f(t)\,dt+{\frac {f(1)+f(x)}{2}}\\&\quad +\sum _{k=2}^{m}{\frac {(-1)^{k}B_{k}}{k!}}{\bigl (}f^{(k-1)}(x)-f^{(k-1)}(1){\bigr )}+R_{m}(x)+C(x),\end{aligned}}}$$where ${\displaystyle B_{k}}$ are the Bernoulli numbers (${\displaystyle B_{1}=-{\tfrac {1}{2}}}$, ${\displaystyle B_{3}=B_{5}=\cdots =0}$), and the remainder term is$${\displaystyle R_{m}(x)=(-1)^{m+1}\int _{1}^{x}{\frac {b_{m}(t)}{m!}}\,f^{(m)}(t)\,dt,}$$with ${\displaystyle b_{m}(t)=B_{m}(t-\lfloor t\rfloor )}$ the periodized Bernoulli polynomial. The terms with odd ${\displaystyle k>1}$ vanish, so the sum effectively runs only over even indices. Choosing ${\displaystyle m=2p}$ gives the form$${\displaystyle {\begin{aligned}\nabla ^{-1}f(x)&=\int _{1}^{x}f(t)\,dt+{\frac {f(1)+f(x)}{2}}\\&\quad +\sum _{k=1}^{p}{\frac {B_{2k}}{(2k)!}}{\bigl (}f^{(2k-1)}(x)-f^{(2k-1)}(1){\bigr )}+R_{2p}(x)+C(x),\end{aligned}}}$$with the remainder$${\displaystyle R_{2p}(x)=-\int _{1}^{x}{\frac {b_{2p}(t)}{(2p)!}}\,f^{(2p)}(t)\,dt.}$$The formula gives the analytic continuation of the discrete sum.