Indefinite sum

In the calculus of finite differences, the indefinite sum (or antidifference operator), denoted by ${\textstyle \sum _{x}}$ or ${\displaystyle \Delta ^{-1}}$,¹² is the linear operator that inverts the forward difference operator ${\displaystyle \Delta f(x)=f(x+1)-f(x).}$ That is, if ${\displaystyle \sum _{x}f(x)=F(x)}$, then ${\displaystyle F}$ satisfies the functional equation

${\displaystyle F(x+1)-F(x)=f(x),}$

so that applying the forward difference recovers the original function:³ ${\displaystyle \Delta \sum _{x}f(x)=f(x).}$ 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 ${\displaystyle C(x)}$ (satisfying ${\displaystyle C(x+1)=C(x)}$), the function ${\displaystyle F(x)+C(x)}$ is also a solution. Therefore, an indefinite sum is unique up to a 1-periodic function ${\displaystyle C(x)}$ instead of up to a constant ${\displaystyle C}$ as the indefinite integral is.

To obtain the unique solution up to a constant ${\displaystyle C}$, 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.⁴ 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.

The inverse forward difference operator, ${\displaystyle \Delta ^{-1}}$ (${\displaystyle F(x+1)-F(x)=f(x)}$), extends the summation up to ${\displaystyle x-1}$, typically starting with the iterator at ${\displaystyle 0}$:

${\displaystyle \,\sum _{k=0}^{x-1}f(k).}$

Some authors analytically extend summation for which the upper limit is the argument without a shift, typically starting the iterator at ${\displaystyle 1}$:⁵⁶⁷

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

In this case, the analytic continuation, ${\displaystyle F(x)}$, for the sum is a solution of ${\displaystyle \nabla ^{-1}f(x)}$. Stated explicitly, that is:

${\displaystyle \ F(x)-F(x-1)=f(x),}$

Which follows from the discrete counterpart:

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

Some authors use the equivalent form called the telescoping equation:⁸

${\displaystyle F(x+1)-F(x)=f(x+1).}$

The lower bounds of the discrete analog for both inverse forward difference and inverse backward difference can be an arbitrary constant other than those listed here, as it is absorbed into the height of the 1-periodic or constant term ${\displaystyle C}$.

Indefinite sums can be used to calculate definite sums with the formula:⁹

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

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

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

The following basic indefinite sums follow from the fundamental properties of the difference operator, where ${\displaystyle C(x)}$ represents an arbitrary 1-periodic function (or a constant if the Nørlund principal solution is assumed):¹⁰

Constant:

${\displaystyle \sum _{x}c=cx+C(x)}$

Exponential:

${\displaystyle \sum _{x}a^{x}={\frac {a^{x}}{a-1}}+C(x)\quad a\neq 1}$

Logarithm:

${\displaystyle \sum _{x}\ln x=\ln \Gamma (x)+C(x)}$

Powers:¹¹

${\displaystyle {\begin{aligned}\sum _{x}x^{a}&={\begin{cases}{\frac {B_{a+1}(x)}{a+1}}+C(x),&{\text{if }}a\neq -1\\\psi (x)+C(x),&{\text{if }}a=-1\end{cases}}\\&={\begin{cases}-\zeta (-a,x)+C(x),&{\text{if }}a\neq -1\\\psi (x)+C(x),&{\text{if }}a=-1\end{cases}}\end{aligned}}}$

where ${\displaystyle B_{a}(x)}$ are the Bernoulli polynomials (via Abel-Plana, Hurwitz zeta, or as defined by their recurrence; not the definition by generating functions), ${\displaystyle \zeta (s,a)}$ is the Hurwitz zeta function, and ${\displaystyle \psi (z)}$ is the digamma function. This is related to the generalized harmonic numbers. Combined with series expansions (such as Taylor series) or partial fraction decomposition, the power formula allows the indefinite summation of many analytic functions and rational functions (term-wise, through the linearity of the operator).

Falling factorials provide the discrete analog of the power rule from differential calculus. In infinitesimal calculus, ${\displaystyle {\frac {d}{dx}}x^{n}=nx^{n-1}}$. In the calculus of finite differences, the falling factorial

${\displaystyle (x)_{n}=x^{\underline {n}}=x(x-1)(x-2)\cdots (x-n+1)={\frac {\Gamma \left(x+1\right)}{\Gamma \left(x-n+1\right)}}}$

plays the role of ${\displaystyle x^{n}}$, and the forward difference operator satisfies

${\displaystyle \Delta (x)_{n}=n(x)_{n-1}.}$

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

${\displaystyle \sum _{x}(x)_{n}={\frac {(x)_{n+1}}{n+1}}+C(x),\quad n\neq -1.}$

Equivalently, using the Gamma function:

${\displaystyle \sum _{x}{\frac {\Gamma \left(x+1\right)}{\Gamma \left(x-n+1\right)}}={\frac {\Gamma \left(x+1\right)}{\left(n+1\right)\Gamma \left(x-n\right)}}+C(x),\quad n\neq -1.}$

For the case where ${\displaystyle n=-1}$, the solution is the digamma function with a shift, ${\displaystyle \psi \left(x+1\right)+C(x)}$, which naturally extends the harmonic numbers.

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 two functions ${\displaystyle u(x)}$ and ${\displaystyle v(x)}$, the product rule for the forward difference is:

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

Introducing the shift operator ${\displaystyle \mathrm {E} }$, defined by ${\displaystyle \mathrm {E} f(x)=f(x+1)}$, this can be written more compactly as:

${\displaystyle \Delta (uv)=u\Delta v+\mathrm {E} v\,\Delta u.}$

Summation by parts. Rearranging the product rule gives:

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

Taking the indefinite sum of both sides and using the fact that ${\displaystyle \sum _{x}\Delta F(x)=F(x)+C(x)}$ (where ${\displaystyle C(x)}$ is an arbitrary 1‑periodic function) yields the formula for summation by parts:¹²¹⁰

${\displaystyle \sum _{x}u(x)\Delta v(x)=u(x)v(x)-\sum _{x}v(x+1)\Delta u(x)+C(x).}$

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

${\displaystyle \sum _{x}f(x)\Delta g(x)+\sum _{x}g(x)\Delta f(x)=f(x)g(x)-\sum _{x}\Delta f(x)\Delta g(x)+C(x).}$

Definite summation by parts. For definite sums from ${\displaystyle a}$ to ${\displaystyle b}$, the formula becomes:

${\displaystyle \sum _{k=a}^{b}u(k)\Delta v(k)={\bigl [}u(b+1)v(b+1)-u(a)v(a){\bigr ]}-\sum _{k=a}^{b}v(k+1)\Delta u(k).}$

The functional equation ${\displaystyle F(x+1)-F(x)=f(x)}$ does not have a unique solution. If ${\displaystyle F_{1}(x)}$ is a particular solution, then for any function ${\displaystyle C(x)}$ satisfying ${\displaystyle C(x+1)=C(x)}$ (i.e., any 1-periodic function), the function ${\displaystyle F_{2}(x)=F_{1}(x)+C(x)}$ is also a solution. Therefore, the indefinite sum operator defines a family of functions differing by an arbitrary 1-periodic component, ${\displaystyle C(x)}$.

To select the unique principal solution (German: Hauptlösung)⁴ up to an additive constant ${\displaystyle C}$ (instead of up to the additive 1-periodic function ${\displaystyle C(x)}$) one must impose additional constraints.

Following the theory developed by Niels Erik Nørlund,⁴ the indefinite sum can be uniquely determined for analytic functions by imposing restriction on their growth in the complex plane. Specifically, by imposing minimal growth, the non-constant periodic terms can be filtered out.

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 connected components. 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 connected half-planes.

Nørlund’s theory provides a principal solution in each connected component 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 connected component's principal solution contains no singularities in its respective connective component, but contains singularities that recur outwards into outer disjoint connected components.

The solution that contains the largest defined portion of the discrete sum being extended is then considered the disjoint connected component defining the canonical principal solution. This usually becomes, in practice, the right half-plane.

Suppose ${\displaystyle f(z)}$ is analytic in a vertical strip containing a segment of the real axis, and let ${\displaystyle F(z)}$ be an analytic solution of ${\displaystyle F(z+1)-F(z)=f(z)}$ in that strip. To ensure uniqueness within that strip, require ${\displaystyle F(z)}$ to be of minimal growth, specifically to be of exponential type less than ${\displaystyle 2\pi }$ in the imaginary direction. That is, there exist constants ${\displaystyle M>0}$ and ${\displaystyle \epsilon >0}$ such that ${\displaystyle |F(z)|\leq Me^{(2\pi -\epsilon )|\Im (z)|}}$ as ${\displaystyle |\Im (z)|\to \infty }$.¹⁴¹⁵

Let ${\displaystyle F_{1}(z)}$ and ${\displaystyle F_{2}(z)}$ be two analytic solutions satisfying this growth condition on the same connected component. Their difference ${\displaystyle C(z)=F_{1}(z)-F_{2}(z)}$ is then analytic, 1-periodic (i.e., ${\displaystyle C(z+1)=C(z)}$), and inherits the same exponential type less than ${\displaystyle 2\pi }$.

Nørlund uses a fundamental result in complex analysis (related to Carlson's theorem, the Phragmén–Lindelöf principle, and the Paley–Wiener theorem) which states that a non-constant periodic entire function must have exponential type at least ${\displaystyle 2\pi }$.⁴ This follows from its Fourier series expansion: if ${\displaystyle C(z)}$ is non-constant, its Fourier series contains a term ${\displaystyle a_{n}e^{2\pi inz}}$ with ${\displaystyle n\neq 0}$, which has type ${\displaystyle 2\pi |n|\geq 2\pi }$. Since ${\displaystyle C(z)}$ has type strictly less than ${\displaystyle 2\pi }$, it cannot contain any such term and therefore must be constant. Hence, on any fixed connected component where the growth condition holds, the solution is unique up to a constant.

The exponential type less than ${\displaystyle 2\pi }$ in the imaginary direction on ${\displaystyle f}$ condition 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).⁴ 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 ${\displaystyle 7>2\pi }$.¹⁶¹⁰

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 denominator¹⁰) the principal solution fails to exist (or is undefined everywhere) because it resonates with the kernel of the difference operator:¹⁷¹⁸¹⁹ $${\displaystyle \Delta ^{-1}={\frac {1}{e^{D}-1}}.}$$ In all other cases (i.e., when ${\displaystyle f}$ is meromorphic and on some vertical strip that contains a segment of the real line and its exponential type is not an integer multiple of ${\displaystyle 2\pi }$) the principal solution exists and is uniquely determined (up to a constant) on that connected component. Different components may give distinct branches; the canonical branch is the one analytic on the component containing the positive integers.

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 ${\displaystyle p}$ 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.²⁰²¹

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 ${\displaystyle \mathbb {R} _{+}}$.

In their paper How to Add a Noninteger Number of Terms,⁵ 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 which 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:

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

Following directly from uniqueness, if ${\displaystyle f(z)}$ is a meromorphic function, one can define a unique analytic solution of the backward difference sum, by imposing the conditions that:

• Difference Equation: ${\displaystyle F(x)-F(x-1)=f(x)}$
• Normalization: ${\displaystyle F(0)=0}$ (empty sum boundary condition).
• Growth constraint: ${\displaystyle F(z)}$ has the minimal possible exponential type in the imaginary direction.

Under these conditions, ${\displaystyle F(z)}$ satisfies a reflection formula (referred to by Nørlund as Ergänzungssatz, a complementary theorem to uniqueness of the principal solution [Hauptlösung], presenting it as ${\displaystyle G\left(x-\omega |-\omega \right)=G\left(x|\omega \right),}$${\displaystyle F\left(x-\omega |-\omega \right)=F\left(x|\omega \right)}$ where ${\displaystyle \omega }$ is the span).¹⁶ From Nørlund’s Ergänzungssatz for the principal solution, one obtains the following symmetry for the inverse backward difference when the summand is odd or even under the condition ${\displaystyle F(0)=0}$ via direct application (setting ${\displaystyle \omega =1}$).

Assume ${\displaystyle f}$ is an odd function (${\displaystyle f(-z)=-f(z)}$) and that a principal solution ${\displaystyle F}$ exists. Define ${\displaystyle H(x)=F(x)-F(-1-x)}$. Using the difference equation ${\displaystyle F(x)-F(x-1)=f(x)}$ and oddness,

${\displaystyle {\begin{aligned}H(x)-H(x-1)&=[F(x)-F(x-1)]-[F(-1-x)-F(-x)]\\&=f(x)-{\bigl [}-f(-x){\bigr ]}=f(x)+f(-x)=0,\end{aligned}}}$

so ${\displaystyle H}$ is 1-periodic. Because ${\displaystyle F}$ has minimal exponential type, ${\displaystyle H}$ does as well; by Nørlund’s uniqueness theorem, a non-constant 1-periodic function of type ${\displaystyle <2\pi }$ must be constant. Hence ${\displaystyle H}$ is constant. Evaluating at ${\displaystyle x=0}$ with ${\displaystyle F(0)=0}$ and ${\displaystyle f(0)=0}$ (oddness) gives ${\displaystyle F(-1)=0}$, so ${\displaystyle H(0)=0}$. Therefore ${\displaystyle H\equiv 0}$, yielding ${\displaystyle 0=F(x)-F(-1-x),}$

${\displaystyle F(z)=F(-1-z),}$

a point symmetry about ${\displaystyle z=-1/2}$. For example, ${\displaystyle f(z)=z}$ gives ${\displaystyle F(z)={\frac {z(z+1)}{2}}}$.¹⁶

If ${\displaystyle f}$ is an even function (${\displaystyle f(-z)=f(z)}$) with a principal solution ${\displaystyle F}$, define ${\displaystyle H(x)=F(x)+F(-1-x)}$. Then

${\displaystyle {\begin{aligned}H(x)-H(x-1)&=[F(x)-F(x-1)]+[F(-1-x)-F(-x)]\\&=f(x)+{\bigl [}-f(-x){\bigr ]}=f(x)-f(-x)=0,\end{aligned}}}$

so again ${\displaystyle H}$ is a constant 1-periodic function. Setting ${\displaystyle x=-1}$ gives the constant: ${\displaystyle H(-1)=F(-1)+F(0)=F(-1)}$. Consequently,

${\displaystyle F(z)+F(-1-z)=F(-1).}$

Because the indefinite sum is defined only up to an arbitrary 1-periodic function, the constant ${\displaystyle C}$ 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.

The most direct method forces the indefinite sum to extend the usual discrete sum and to satisfy the empty sum convention. This is the same as ${\displaystyle \lim _{x\to 0^{-}}F(x)=0\ {\text{or}}\ \lim _{x\to 0^{+}}F(x)=0.}$

Inverse backward difference

${\displaystyle \nabla ^{-1}f(x)}$ corresponds to ${\displaystyle \textstyle \sum _{k=1}^{x}f(k)}$. The convention ${\displaystyle \left.\nabla ^{-1}f(x)\right|_{x=0}=0}$ makes the sum over an empty interval zero.⁵¹⁰

Inverse forward difference

${\displaystyle \Delta ^{-1}f(x)}$ corresponds to ${\displaystyle \textstyle \sum _{k=0}^{x-1}f(k)}$. The same convention yields ${\displaystyle \left.\Delta ^{-1}f(x)\right|_{x=0}=0}$.

These conditions determine the solution uniquely up to an additive constant. For example,¹¹

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

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

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 Bernoulli polynomials ${\displaystyle B_{n}(x)}$ are defined by the generating function

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

together with the normalization

${\displaystyle \int _{0}^{1}B_{n}(x)\,dx=0\qquad (n\geq 1).}$

This property follows from the difference equation ${\displaystyle \Delta B_{\nu }(x)=\nu x^{\nu -1}}$ and the integration formula ${\displaystyle \int _{x}^{x+1}B_{\nu }(z)\,dz=x^{\nu }}$, derived by Nørlund²² and found in standard references.

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 context of Bernoulli polynomials, the Hurwitz or Riemann zeta functions, generalized harmonic number function, or when dealing with monomials.

In the symbolic method developed by Niels Erik Nørlund and L. M. Milne-Thomson, the indefinite product operator ${\displaystyle \prod _{x}}$ serves as the multiplicative analog to the indefinite sum. It is defined by the first order homogeneous equation ${\displaystyle F(x+1)=f(x)F(x).}$

By taking the logarithm of the product formula, one obtains the telescoping identity ${\displaystyle \Delta \ln F(x)=\ln f(x)}$.²³ This allows the indefinite product to be expressed through an indefinite sum:

${\displaystyle \prod _{x}f(x)=\varpi (x)\exp \left(\sum _{x}\ln f(x)\right),}$

where ${\displaystyle \varpi (x)}$ is an arbitrary periodic function of period 1.²⁴ This representation is valid provided a branch of the logarithm can be chosen so that ${\displaystyle \ln \left(f\left(x\right)\right)}$ is single-valued and its indefinite sum exists. Conversely, an indefinite sum may be represented as the logarithm of an indefinite product:

${\displaystyle \sum _{x}f(x)=\ln \left(\prod _{x}\exp(f(x))\right)+C(x).}$

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:⁷ $${\displaystyle {\begin{aligned}\nabla ^{-1}f(x)&=\int _{1}^{x}f(t)dt+{\frac {f(1)+f(x)}{2}}\\&\quad +i\int _{0}^{\infty }{\frac {\left(f(x-it)-f(1-it)\right)-\left(f(x+it)-f(1+it)\right)}{e^{2\pi t}-1}}dt+C(x)\end{aligned}}}$$

This analytic continuation is valid when the conditions for the original formula are met. The sufficient conditions are:¹⁴¹⁵

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 ${\displaystyle |\Im (z)|\to \infty .}$

For an entire function of exponential type less than ${\displaystyle \ln \left(2\right)}$²⁵ the inverse forward difference operator, ${\displaystyle \Delta ^{-1}f(x)}$, can be expressed by its Newton series expansion: ²⁶²⁷

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

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

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:¹⁷¹⁸¹⁹

$${\displaystyle \Delta ^{-1}={\frac {1}{e^{D}-1}}=\sum _{v=0}^{\infty }{\frac {B_{v}}{v!}}D^{\,v-1},}$$

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:¹⁸

$${\displaystyle \sum _{x}x^{n}={\frac {B_{n+1}(x)}{n+1}}+C(x),}$$

where ${\displaystyle B_{n}(x)}$ are the Bernoulli polynomials of the first order.¹⁸

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:¹⁹

$${\displaystyle \sum _{x}f(x)=\sum _{n=1}^{\infty }{\frac {f^{(n-1)}(0)}{n!}}B_{n}(x)+C(x).}$$

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 to the same expansion, but with ${\displaystyle B_{1}=+{\tfrac {1}{2}}}$ in place of ${\displaystyle B_{1}=-{\tfrac {1}{2}}}$.

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:⁶¹⁴

$${\displaystyle {\begin{aligned}\nabla ^{-1}f(x)&=\int _{1}^{x}f(t)\,dt+{\frac {f(1)+f(x)}{2}}\\&\quad +\sum _{k=2}^{m}{\frac {(-1)^{k}B_{k}}{k!}}{\bigl (}f^{(k-1)}(x)-f^{(k-1)}(1){\bigr )}+R_{m}(x)+C(x),\end{aligned}}}$$

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.

Laplace's summation formula, closely related to the Gregory summation formula, can be seen as the discrete counterpart to the Euler–Maclaurin formula. The inverse forward difference ${\displaystyle \Delta ^{-1}f(x)}$:^28291330

${\displaystyle \sum _{x}f(x)=\int _{0}^{x}f(t)dt-\sum _{k=1}^{\infty }{\frac {c_{k}}{k!}}\Delta ^{k-1}f(x)+C(x)}$

where ${\displaystyle c_{k}=\int _{0}^{1}(x)_{k}dx}$ are the Cauchy numbers of the first kind.

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

Truncating the series after ${\displaystyle n}$ terms leaves a remainder that can be expressed as an integral of ${\displaystyle f^{(n)}}$ times a periodic Bernoulli polynomial.¹³³⁰ In the notation of Charles Jordan, Gregory's formula is:¹³

$${\displaystyle {\begin{aligned}\sum _{x=a}^{z}f(x)&=\int _{a}^{z}f(x)\,dx\\&\quad -\sum _{m=1}^{n}b_{m}{\bigl [}\Delta ^{m-1}f(z)-\Delta ^{m-1}f(a){\bigr ]}\\&\quad -b_{n}\,(z-a)\,\Delta ^{n}f(\xi ),\quad a<\xi <z,\end{aligned}}}$$ where the coefficients ${\displaystyle b_{m}}$ are the Bernoulli numbers of the second kind. Note the argument is without a shift, aligning with the inverse backward difference.

See also

See also

• Indefinite product
• Time scale calculus
• List of derivatives and integrals in alternative calculi

References

References

1. Man, Yiu-Kwong (1993), "On computing closed forms for indefinite summations", Journal of Symbolic Computation, 16 (4): 355–376, doi:10.1006/jsco.1993.1053, MR 1263873
2. Goldberg, Samuel (1986) [1958]. Introduction to Difference Equations, with Illustrative Examples from Economics, Psychology, and Sociology. New York: Dover Publications. p. 41. ISBN 978-0-486-65084-5. MR 0094249. If ${\displaystyle Y}$ is a function whose first difference is the function ${\displaystyle y}$, then ${\displaystyle Y}$ is called an indefinite sum of ${\displaystyle y}$ and denoted by ${\displaystyle \Delta ^{-1}y}$.
3. Kelley, Walter G.; Peterson, Allan C. (2001). Difference Equations: An Introduction with Applications. Academic Press. p. 20. ISBN 0-12-403330-X.
4. Nörlund, Niels Erik. Vorlesungen über Differenzenrechnung. Springer. pp. 40–44. ISBN 978-3-642-50514-0.
5. Markus Müller and Dierk Schleicher, How to Add a Noninteger Number of Terms: From Axioms to New Identities, Amer. Math. Mon. 118(2), 136-152 (2011).
6. Candelpergher, Bernard (2017). "Ramanujan Summation of Divergent Series" (PDF). HAL Archives Ouvertes. p. 3. Retrieved 2025-12-07.
7. Candelpergher, Bernard (2017). "Ramanujan Summation of Divergent Series" (PDF). HAL Archives Ouvertes. p. 23. Retrieved 2025-12-07.
8. Algorithms for Nonlinear Higher Order Difference Equations, Manuel Kauers
9. "Handbook of discrete and combinatorial mathematics", Kenneth H. Rosen, John G. Michaels, CRC Press, 1999, ISBN 0-8493-0149-1
10. Jordan, Charles (1960). Calculus of Finite Differences (Second ed.). New York, NY: Chelsea Publishing Company. pp. 104–107.
11. Candelpergher, Bernard (2017). "Ramanujan Summation of Divergent Series" (PDF). HAL Archives Ouvertes. pp. 18–23.
12. Kelley, Walter G.; Peterson, Allan C. (2001). Difference Equations: An Introduction with Applications. Academic Press. p. 24. ISBN 0-12-403330-X.
13. Jordan, Charles (1960). Calculus of Finite Differences (Second ed.). New York, NY: Chelsea Publishing Company. pp. 284–285.
14. "§2.10 Sums and Sequences". NIST Digital Library of Mathematical Functions. National Institute of Standards and Technology. Retrieved 2025-11-20.
15. Olver, Frank W. J. (1997). Asymptotics and Special Functions. A K Peters Ltd. p. 290. ISBN 978-1-56881-069-0.
16. Nörlund, Niels Erik. Vorlesungen über Differenzenrechnung. Springer. pp. 73–74. ISBN 978-3-642-50514-0.
17. Steffensen, J. F. (1950). Interpolation (2nd ed.). New York, NY: Chelsea Publishing Company. p. 192.
18. Milne-Thomson, L. M. (1933). The Calculus of Finite Differences. Macmillan and Co. pp. 139–140.
19. Nörlund, Niels Erik. Vorlesungen über Differenzenrechnung. Springer. pp. 142–143. ISBN 978-3-642-50514-0.
20. Marichal, Jean‑Luc; Zenaïdi, Naïm (2024). "A generalization of Bohr‑Mollerup's theorem for higher order convex functions: a tutorial". Aequationes Mathematicae. 98 (2): 455–481. arXiv:2207.12694. doi:10.1007/s00010-023-00968-9.
21. Marichal, Jean‑Luc; Zenaïdi, Naïm (2022). A Generalization of Bohr‑Mollerup's Theorem for Higher Order Convex Functions. Developments in Mathematics. Vol. 70. Springer. doi:10.1007/978-3-030-95088-0. ISBN 978-3-030-95087-3.
22. Nörlund, Niels Erik. Vorlesungen über Differenzenrechnung. Springer. p. 19. ISBN 978-3-642-50514-0.
23. Nörlund, Niels Erik. Vorlesungen über Differenzenrechnung. Springer. p. 109. ISBN 978-3-642-50514-0.
24. Milne-Thomson, L. M. (1933). The Calculus of Finite Differences. Macmillan and Co. pp. 324–325.
25. Nörlund, Niels Erik. Vorlesungen über Differenzenrechnung. Springer. p. 237. ISBN 978-3-642-50514-0.
26. Newton, Isaac, (1687). Principia, Book III, Lemma V, Case 1
27. Iaroslav V. Blagouchine (2018). "Three notes on Ser's and Hasse's representations for the zeta-functions" (PDF). Integers (Electronic Journal of Combinatorial Number Theory). 18A: 1–45. arXiv:1606.02044. doi:10.5281/zenodo.10581385.
28. Bernoulli numbers of the second kind on Mathworld
29. Ferraro, Giovanni (2008). The Rise and Development of the Theory of Series up to the Early 1820s. Springer Science+Business Media, LLC. p. 248. ISBN 978-0-387-73468-2.
30. Milne-Thomson, L. M. (1933). The Calculus of Finite Differences. Macmillan and Co. pp. 180–181.

Further reading

Further reading

• "Difference Equations: An Introduction with Applications", Walter G. Kelley, Allan C. Peterson, Academic Press, 2001, ISBN 0-12-403330-X
• Markus Müller. How to Add a Non-Integer Number of Terms, and How to Produce Unusual Infinite Summations
• Markus Mueller, Dierk Schleicher. Fractional Sums and Euler-like Identities
• S. P. Polyakov. Indefinite summation of rational functions with additional minimization of the summable part. Programmirovanie, 2008, Vol. 34, No. 2.
• "Finite-Difference Equations And Simulations", Francis B. Hildebrand, Prenctice-Hall, 1968

External links

External links

• Brian Hamrick: Discrete Calculus (PDF, 70 kB)
• Interactive visualization of the Nörlund principal solution for inverse backward differences. Implements Candelpergher's analytic continuation (Abel-Plana formula with recurrence) for visualizing Nörlund's principal solution.