Indefinite sum

In the calculus of finite differences, the indefinite sum operator (also known as the antidifference operator), denoted by $\sum _x$ or ${\displaystyle {\mathrm{\Delta }}^{-1}}$,^[1][2] is the linear operator that is the inverse of the forward difference operator, ${\displaystyle \mathrm{\Delta }}$:

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

It relates to the forward difference operator as the indefinite integral relates to the derivative. In the same way that the indefinite integral solves for the family of functions that differentiate to ${\displaystyle f}$, the indefinite sum solves for which family of functions have ${\displaystyle f}$ as their forward difference.

If ${\displaystyle \sum _{x}f(x)=F(x)}$, then ${\displaystyle F}$ satisfies the functional equation

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

Applying the forward difference operator to an indefinite sum returns the original function:^[3]

${\displaystyle \mathrm{\Delta }\sum _{x}f(x)=f(x).}$

In the notation ${\displaystyle \sum _{x}f(x)}$, the variable ${\displaystyle x}$ plays the same role as the index variable in a discrete sum; it indicates the argument at which the antidifference is to be evaluated. The subscript ${\displaystyle x}$ acts as a placeholder, analogous to the ${\displaystyle k}$ in ${\displaystyle {\displaystyle \sum _{k=0}^{x-1}}f(k)}$, and specifies that the antidifference ${\displaystyle F}$ is a function of ${\displaystyle x}$.

The solution ${\displaystyle F(x)}$ is not unique: if ${\displaystyle F(x)}$ is one solution, then for any 1‑periodic function ${\displaystyle C(x)}$ (i.e., ${\displaystyle C(x+1)=C(x)}$), 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 constraints. The Nørlund principal solution is the unique analytic solution that has the minimal possible exponential type, filtering out any non‑constant periodic component.^[4]

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

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

Some authors analytically extend summation for which the upper limit is the argument without a shift, typically starting the iterator at ${\displaystyle 1}$:^[5][6][7]

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

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

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

Which follows from the discrete counterpart:

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

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]

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

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

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

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:

${\displaystyle \sum _x{a}^x=\frac{a^x}{a-1}+C(x)a\ne 1}$

Logarithm:

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

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

${\displaystyle (x{)}_n=x^{\underset{\_}{n}}=x(x-1)(x-2)\cdots (x-n+1)=\frac{\mathrm{\Gamma }(x+1)}{\mathrm{\Gamma }(x-n+1)}}$

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

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

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

${\displaystyle \sum _x(x{)}_n=\frac{(x{)}_{n+1}}{n+1}+C(x),n\ne -1.}$

Equivalently, using the Gamma function:

${\displaystyle \sum _x\frac{\mathrm{\Gamma }(x+1)}{\mathrm{\Gamma }(x-n+1)}=\frac{\mathrm{\Gamma }(x+1)}{(n+1)\mathrm{\Gamma }(x-n)}+C(x),n\ne -1.}$

For the case where ${\displaystyle n=-1}$, the solution is the digamma function with a shift, ${\displaystyle \psi (x+1)+C(x)}$, 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:

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

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

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

Summation by parts. Rearranging the product rule gives:

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

Taking the indefinite sum of both sides and using the fact that ${\displaystyle \sum _x\mathrm{\Delta }F(x)=F(x)+C(x)}$ (where ${\displaystyle C(x)}$ is an arbitrary 1‑periodic function) yields the formula for summation by parts:^[11][10]

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

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

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

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

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

File:Finite Difference Visualisation One Periodic Vanishing.webm 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.

Suppose ${\displaystyle f(z)}$ is analytic in a vertical strip containing 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, 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 ${\displaystyle ϵ>0}$ such that ${\displaystyle |F(z)|\le M{e}^{(2\pi -ϵ)|\mathrm{\Im }(z)|}}$ as ${\displaystyle |\mathrm{\Im }(z)|\to \mathrm{\infty }}$.^[13][14]

Let ${\displaystyle F_1(z)}$ and ${\displaystyle F_2(z)}$ be two analytic solutions satisfying this growth condition. 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 ${\displaystyle n\ne 0}$, which has type ${\displaystyle 2\pi |n|\ge 2\pi }$. Since ${\displaystyle C(z)}$ has type strictly less than ${\displaystyle 2\pi }$, it cannot contain any such term and therefore must be constant.

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 the minimal possible exponential type for the given ${\displaystyle f}$.^[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 ${\displaystyle 7>2\pi }$.

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 (or is undefined everywhere) because it resonates with the kernel of the difference operator. In all other cases (i.e., when ${\displaystyle f}$ is meromorphic and its exponential type in some vertical strip is not an integer multiple of ${\displaystyle 2\pi }$) the principal solution exists and is uniquely determined by minimal exponential type.

In real analysis, the uniqueness condition can be given using higher‑order convexity, generalizing the Bohr-Mollerup theorem. For an integer ${\displaystyle p\ge 0}$, 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 ${\displaystyle (M,\mathrm{\infty })}$.

Marichal and Zenaïdi proved the following uniqueness theorem, their method requiring the solution to be eventually ${\displaystyle p}$-convex or ${\displaystyle p}$-concave.^[15][16]

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

${\displaystyle f(x)=f(1)+\underset{n\to \mathrm{\infty }}{\lim }({\displaystyle \sum _{k=1}^{n-1}}g(k)-{\displaystyle \sum _{k=0}^{n-1}}g(x+k)+{\displaystyle \sum _{j=1}^p}{\displaystyle (\genfrac{}{}{0ex}{}{x}{j})}{\mathrm{\Delta }}^{j-1}g(n)),}$

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

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

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

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

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

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 ${\displaystyle \sigma \in \mathbb{\mathbb{N}}\cup \{-\mathrm{\infty }\}}$. A function ${\displaystyle f:U\to \mathbb{ℂ}}$ 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 ${\displaystyle (p_n{)}_{n\in \mathbb{\mathbb{N}}}}$ of fixed degree ${\displaystyle \sigma }$ such that for all ${\displaystyle x\in U}$

${\displaystyle |f(n+x)-p_n(n+x)|\to 0}$ as ${\displaystyle n\to +\mathrm{\infty }}$

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

${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }({\displaystyle \sum _{\nu =n+x}^{n+y}}{p}_n(\nu )+{\displaystyle \sum _{\nu =1}^n}(f(\nu +x-1)-f(\nu +y)\left)\right),}$

exists.

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

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

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(x-\omega |-\omega )=G(x|\omega ),}$${\displaystyle F(x-\omega |-\omega )=F(x|\omega )}$ where ${\displaystyle \omega }$ is the span).^[17]

If ${\displaystyle f(z)}$ is an odd function (${\displaystyle f(-z)=-f(z)}$), the unique analytic solution ${\displaystyle F(z)}$ satisfies:^[17]

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

This represents a point symmetry about ${\displaystyle z=-1/2}$.

If ${\displaystyle f(z)}$ is an even function (${\displaystyle f(-z)=f(z)}$), the unique analytic solution ${\displaystyle F(z)}$ satisfies:^[17]

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

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 ${\displaystyle \mathrm{\Delta }\ln F(x)=\ln f(x)}$.^[18] This allows any indefinite product to be expressed through an indefinite sum:

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

where ${\displaystyle \varpi (x)}$ is an arbitrary periodic function of period 1.^[19] Conversely, an indefinite sum may be represented as the logarithm of an indefinite product:

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

For an entire function of exponential type less than ${\displaystyle \ln \left(2\right)}$^[20] the inverse forward difference operator, ${\displaystyle {\mathrm{\Delta }}^{-1}f(x)}$, can be expressed by its Newton series expansion: ^[21][22]

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

${\displaystyle (x{)}_k=\frac{\mathrm{\Gamma }(x+1)}{\mathrm{\Gamma }(x-k+1)}}$ 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:^[23][24][25]

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

where ${\displaystyle B_v}$ are the Bernoulli numbers defined by the generating function ${\displaystyle \frac{t}{e^t-1}={\displaystyle \sum _{v=0}^{\mathrm{\infty }}}{B}_v\frac{t^v}{v!}}$. Under this convention ${\displaystyle B_1=-\frac{1}{2}}$.

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:^[24]

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

where ${\displaystyle B_n(x)}$ are the Bernoulli polynomials of the first order.^[24]

If ${\displaystyle f}$ admits a Maclaurin series expansion ${\displaystyle f(x)={\displaystyle \sum _{n=0}^{\mathrm{\infty }}}\frac{f^{(n)}(0)}{n!}{x}^n}$, the antidifference of monomials in the series expansion yields the formal series:^[25]

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

For non‑polynomials this expansion is generally asymptotic.

Relation to the inverse backward difference

If one instead expands the inverse backward difference operator, ${\displaystyle {\mathrm{\nabla }}^{-1}=\frac{e^D}{e^D-1}}$ (which extends ${\displaystyle {\displaystyle \sum _{k=1}^x}f(k)}$), it admits to the same expansion, but with ${\displaystyle B_1=+\frac{1}{2}}$ in place of ${\displaystyle B_1=-\frac{1}{2}}$.

The Euler–Maclaurin formula extends ${\displaystyle {\mathrm{\nabla }}^{-1}f(x)={\displaystyle \sum _{k=1}^x}f(k)}$:^[6][13] $${\displaystyle \begin{array}{rl}{\mathrm{\nabla }}^{-1}f(x)& ={\displaystyle \underset{1}{\overset{x}{\int }}}f(t)dt+\frac{f(1)+f(x)}{2}\\ & +{\displaystyle \sum _{k=1}^p}\frac{B_{2k}}{(2k)!}(f^{(2k-1)}(x)-f^{(2k-1)}(1))+R_p+C(x)\end{array}}$$ where ${\displaystyle B_{2k}}$ are the even Bernoulli numbers, ${\displaystyle p}$ is an arbitrary positive integer, and ${\displaystyle R_p}$ is the remainder term given by:

${\displaystyle R_p=(-1{)}^{p+1}{\displaystyle \underset{1}{\overset{x}{\int }}}{f}^{(p)}(t)\frac{P_p(t)}{p!}dt,}$

with ${\displaystyle P_p(t)=B_p(t-\lfloor t\rfloor )}$ being the periodized Bernoulli function related to the Bernoulli polynomials.

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 ${\displaystyle {\mathrm{\Delta }}^{-1}f(x)}$:^{[26][27][12][28]}

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

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

${\displaystyle (x{)}_k=\frac{\mathrm{\Gamma }(x+1)}{\mathrm{\Gamma }(x-k+1)}}$ 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.^[12][28] In the notation of Charles Jordan, Gregory's formula is:^[12]

$${\displaystyle \begin{array}{rl}{\displaystyle \sum _{x=a}^z}f(x)& ={\displaystyle \underset{a}{\overset{z}{\int }}}f(x)dx\\ & -{\displaystyle \sum _{m=1}^n}{b}_m[{\mathrm{\Delta }}^{m-1}f(z)-{\mathrm{\Delta }}^{m-1}f(a)]\\ & -b_n(z-a){\mathrm{\Delta }}^{n}f(\xi ),a<\xi <z,\end{array}}$$ 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.

The indefinite sum ${\displaystyle {\mathrm{\nabla }}^{-1}f(x)={\displaystyle \sum _{k=1}^x}f(k)}$ can be analytically continued by applying the standard Abel-Plana formula to the finite sum ${\displaystyle {\displaystyle \sum _{k=1}^n}f(k)}$ and then analytically continuing the integer limit ${\displaystyle n}$ to the variable ${\displaystyle x}$. This yields the formula:^[7] $${\displaystyle \begin{array}{rl}{\mathrm{\nabla }}^{-1}f(x)& ={\displaystyle \underset{1}{\overset{x}{\int }}}f(t)dt+\frac{f(1)+f(x)}{2}\\ & +i{\displaystyle \underset{0}{\overset{\mathrm{\infty }}{\int }}}\frac{(f(x-it)-f(1-it))-(f(x+it)-f(1+it))}{e^{2\pi t}-1}dt+C(x)\end{array}}$$

This analytic continuation is valid when the conditions for the original formula are met. The sufficient conditions are:^[13][14]

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

The constant ${\displaystyle C}$ is often fixed using integral conditions, which is consistent with Bernoulli polynomials.

Let ${\displaystyle F(x)={\mathrm{\nabla }}^{-1}f(x)+C}$. Then, the constant ${\displaystyle C}$ is fixed from the condition ${\displaystyle {\displaystyle \underset{-1}{\overset{0}{\int }}}F(x)dx=0}$ or ${\displaystyle {\displaystyle \underset{0}{\overset{1}{\int }}}F(x)dx=0}$.

For example, ${\displaystyle {\displaystyle \underset{-1}{\overset{0}{\int }}}\frac{x(x+1)}{2}dx=-\frac{1}{12}}$ where ${\displaystyle {\mathrm{\nabla }}^{-1}x=\frac{x(x+1)}{2}+C(x).}$

Let ${\displaystyle F(x)={\mathrm{\Delta }}^{-1}f(x)+C}$. Then, the constant ${\displaystyle C}$ is fixed from the condition ${\displaystyle {\displaystyle \underset{0}{\overset{1}{\int }}}F(x)dx=0}$ or ${\displaystyle {\displaystyle \underset{1}{\overset{2}{\int }}}F(x)dx=0.}$

Alternatively, Ramanujan summation can be used: ${\displaystyle {\displaystyle \sum _{x\ge 1}^{\mathrm{\Re }}}f(x)=-f(0)-F(0).}$ Or at 1: ${\displaystyle {\displaystyle \sum _{x\ge 1}^{\mathrm{\Re }}}f(x)=-F(1)}$ respectively.^[29][30]

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

• "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

• 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.

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