Indefinite sum

In discrete calculus, the indefinite sum operator (also known as the antidifference operator), denoted by ${\textstyle \sum _{x}}$ or ${\displaystyle \Delta ^{-1}}$,^[1][2] is the linear operator that is the inverse of the forward difference operator ${\displaystyle \Delta }$. It relates to the forward difference operator as the indefinite integral relates to the derivative. Thus

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

More explicitly, if ${\textstyle \sum _{x}f(x)=F(x)}$, then

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

If F(x) is a solution of this functional equation for a given f(x), then so is F(x)+C(x) for any periodic function C(x) with period 1. Therefore, each indefinite sum actually represents a family of functions. However, due to the Carlson's theorem, the solution equal to its Newton series expansion is unique up to an additive constant C. This unique solution can be represented by formal power series form of the antidifference operator: ${\displaystyle \Delta ^{-1}={\frac {1}{e^{D}-1}}}$.

Fundamental theorem of discrete calculus

Indefinite sums can be used to calculate definite sums with the formula^[3]

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

Definitions

Laplace summation formula

The Laplace summation formula allows the indefinite sum to be written as the indefinite integral plus correction terms obtained from iterating the difference operator, although it was originally developed for the reverse process of writing an integral as an indefinite sum plus correction terms. As usual with indefinite sums and indefinite integrals, it is valid up to an arbitrary choice of the constant of integration. Using operator algebra avoids cluttering the formula with repeated copies of the function to be operated on:^[4]$${\displaystyle \sum _{x}=\int {}+{\frac {1}{2}}-{\frac {1}{12}}\Delta +{\frac {1}{24}}\Delta ^{2}-{\frac {19}{720}}\Delta ^{3}+{\frac {3}{160}}\Delta ^{4}-\cdots }$$In this formula, for instance, the term ${\displaystyle {\tfrac {1}{2}}}$ represents an operator that divides the given function by two. The coefficients ${\displaystyle +{\tfrac {1}{2}},-{\tfrac {1}{12}},}$ etc., appearing in this formula are the Gregory coefficients, also called Laplace numbers. The coefficient in the term ${\displaystyle \Delta ^{n-1}}$ is^[4]$${\displaystyle {\frac {{\mathcal {C}}_{n}}{n!}}=\int _{0}^{1}{\binom {x}{n}}\,dx,}$$where the numerator ${\displaystyle {\mathcal {C}}_{n}}$ of the left hand side is called a Cauchy number of the first kind, although this name sometimes applies to the Gregory coefficients themselves.^[4]

Newton's formula

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

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

Faulhaber's formula

Main article: Faulhaber's formula

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

Faulhaber's formula provides that the right-hand side of the equation converges.

Mueller's formula

If ${\displaystyle \lim _{x\to {+\infty }}f(x)=0,}$ then^[5]

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

Euler–Maclaurin formula

Main article: Euler–Maclaurin formula

${\displaystyle \sum _{x}f(x)=\int _{0}^{x}f(t)dt-{\frac {1}{2}}f(x)+\sum _{k=1}^{\infty }{\frac {B_{2k}}{(2k)!}}f^{(2k-1)}(x)+C}$

Choice of the constant term

Often the constant C in indefinite sum is fixed from the following condition.

Let

${\displaystyle F(x)=\sum _{x}f(x)+C}$

Then the constant C is fixed from the condition

${\displaystyle \int _{0}^{1}F(x)\,dx=0}$

or

${\displaystyle \int _{1}^{2}F(x)\,dx=0}$

Alternatively, Ramanujan's sum can be used:

${\displaystyle \sum _{x\geq 1}^{\Re }f(x)=-f(0)-F(0)}$

or at 1

${\displaystyle \sum _{x\geq 1}^{\Re }f(x)=-F(1)}$

respectively.^[6][7]

Summation by parts

Main article: Summation by parts

Indefinite summation by parts:

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

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

Definite summation by parts:

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

Period rules

If ${\displaystyle T}$ is a period of function ${\displaystyle f(x)}$ then

${\displaystyle \sum _{x}f(Tx)=xf(Tx)+C.}$

If ${\displaystyle T}$ is an antiperiod of function ${\displaystyle f(x)}$, that is ${\displaystyle f(x+T)=-f(x)}$ then

${\displaystyle \sum _{x}f(Tx)=-{\frac {1}{2}}f(Tx)+C.}$

Alternative usage

Some authors use the phrase "indefinite sum" to describe a sum in which the numerical value of the upper limit is not given:

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

In this case, a closed-form expression F(x) for the sum is a solution of

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

which is called the telescoping equation.^[8] It is the inverse of the backward difference operator ${\displaystyle \nabla }$, ${\displaystyle \nabla ^{-1}}$:

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

It is related to the forward antidifference operator using the fundamental theorem of discrete calculus described earlier.

List of indefinite sums

Antidifferences of rational functions

For positive integer exponents, Faulhaber's formula can be used. Note that ${\displaystyle x}$ in the result of Faulhaber's formula must be replaced with ${\displaystyle x-1}$ due to the offset, as Faulhauber's formula finds ${\displaystyle \nabla ^{-1}}$ rather than ${\displaystyle \Delta ^{-1}}$.

For negative integer exponents, the indefinite sum is closely related to the polygamma function:

${\displaystyle \sum _{x}{\frac {1}{x^{a}}}={\frac {(-1)^{a-1}\psi ^{(a-1)}(x)}{(a-1)!}}+C,\,a\in \mathbb {N} }$

For fractions not listed in this section, one may use the polygamma function with partial fraction decomposition. More generally,

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

where ${\displaystyle B_{a}(x)}$ are the Bernoulli polynomials, ${\displaystyle \zeta (s,a)}$ is the Hurwitz zeta function, and ${\displaystyle \psi (z)}$ is the digamma function. This is related to the generalized harmonic numbers.

As the generalized harmonic numbers use reciprocal powers, ${\displaystyle a}$ must be substituted for ${\displaystyle -a}$, and the most common form uses the inverse of the backward difference offset:

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

Here, ${\displaystyle \zeta (-a)}$ is the constant ${\displaystyle C}$.

The Bernoulli polynomials are also related via a partial derivative with respect to ${\displaystyle x}$:

${\displaystyle {\frac {\partial }{\partial x}}\left(\sum _{x}x^{a}\right)=B_{a}(x)=-a\zeta (1-a,x).}$

Similarly, using the inverse of the backwards difference operator may be considered more natural, as:

${\displaystyle {\frac {\partial }{\partial x}}\left(\nabla ^{-1}x^{a}\right){\bigg |}_{x=0}=-a\zeta (1-a,x+1){\bigg |}_{x=0}=-a\zeta (1-a)=B_{a}.}$

Further generalization comes from use of the Lerch transcendent:

${\displaystyle \sum _{x}{\frac {z^{x}}{(x+a)^{s}}}=-z^{x}\,\Phi (z,s,x+a)+C,}$

which generalizes the generalized harmonic numbers as ${\displaystyle z\Phi \left(z,s,a+1\right)-z^{x+1}\Phi \left(z,s,x+1+a\right)}$ when taking ${\displaystyle \nabla ^{-1}}$. Additionally, the partial derivative is given by

${\displaystyle {\frac {\partial }{\partial x}}\left(-z^{x}\Phi \left(z,s,x+a\right)\right)=z^{x}\left(s\Phi \left(z,s+1,x+a\right)-\ln(z)\Phi \left(z,s,x+a\right)\right).}$

For further information, refer to balanced polygamma function, which provides an alternative to polygamma with nicer analytic properties and Hurwitz zeta function#Special cases and generalizations.

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

Antidifferences of exponential functions

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

Antidifferences of logarithmic functions

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

${\displaystyle \sum _{x}\log _{b}ax=\log _{b}(a^{x-1}\Gamma (x))+C}$

Antidifferences of hyperbolic functions

${\displaystyle \sum _{x}\sinh ax={\frac {1}{2}}\operatorname {csch} \left({\frac {a}{2}}\right)\cosh \left({\frac {a}{2}}-ax\right)+C}$

${\displaystyle \sum _{x}\cosh ax={\frac {1}{2}}\operatorname {csch} \left({\frac {a}{2}}\right)\sinh \left(ax-{\frac {a}{2}}\right)+C}$

${\displaystyle \sum _{x}\tanh ax={\frac {1}{a}}\psi _{e^{a}}\left(x-{\frac {i\pi }{2a}}\right)+{\frac {1}{a}}\psi _{e^{a}}\left(x+{\frac {i\pi }{2a}}\right)-x+C}$

where ${\displaystyle \psi _{q}(x)}$ is the q-digamma function.

Antidifferences of trigonometric functions

${\displaystyle \sum _{x}\sin ax=-{\frac {1}{2}}\csc \left({\frac {a}{2}}\right)\cos \left({\frac {a}{2}}-ax\right)+C\,,\,\,a\neq 2n\pi }$

${\displaystyle \sum _{x}\cos ax={\frac {1}{2}}\csc \left({\frac {a}{2}}\right)\sin \left(ax-{\frac {a}{2}}\right)+C\,,\,\,a\neq 2n\pi }$

${\displaystyle \sum _{x}\sin ^{2}ax={\frac {x}{2}}+{\frac {1}{4}}\csc(a)\sin(a-2ax)+C\,\,,\,\,a\neq n\pi }$

${\displaystyle \sum _{x}\cos ^{2}ax={\frac {x}{2}}-{\frac {1}{4}}\csc(a)\sin(a-2ax)+C\,\,,\,\,a\neq n\pi }$

${\displaystyle \sum _{x}\tan ax=ix-{\frac {1}{a}}\psi _{e^{2ia}}\left(x-{\frac {\pi }{2a}}\right)+C\,,\,\,a\neq {\frac {n\pi }{2}}}$

where ${\displaystyle \psi _{q}(x)}$ is the q-digamma function.

${\displaystyle \sum _{x}\tan x=ix-\psi _{e^{2i}}\left(x+{\frac {\pi }{2}}\right)+C=-\sum _{k=1}^{\infty }\left(\psi \left(k\pi -{\frac {\pi }{2}}+1-x\right)+\psi \left(k\pi -{\frac {\pi }{2}}+x\right)-\psi \left(k\pi -{\frac {\pi }{2}}+1\right)-\psi \left(k\pi -{\frac {\pi }{2}}\right)\right)+C}$

${\displaystyle \sum _{x}\cot ax=-ix-{\frac {i\psi _{e^{2ia}}(x)}{a}}+C\,,\,\,a\neq {\frac {n\pi }{2}}}$

${\displaystyle \sum _{x}\operatorname {sinc} x=\operatorname {sinc} (x-1)\left({\frac {1}{2}}+(x-1)\left(\ln(2)+{\frac {\psi ({\frac {x-1}{2}})+\psi ({\frac {1-x}{2}})}{2}}-{\frac {\psi (x-1)+\psi (1-x)}{2}}\right)\right)+C}$

where ${\displaystyle \operatorname {sinc} (x)}$ is the normalized sinc function.

Antidifferences of inverse hyperbolic functions

${\displaystyle \sum _{x}\operatorname {artanh} \,ax={\frac {1}{2}}\ln \left({\frac {\Gamma \left(x+{\frac {1}{a}}\right)}{\Gamma \left(x-{\frac {1}{a}}\right)}}\right)+C}$

Antidifferences of inverse trigonometric functions

${\displaystyle \sum _{x}\arctan ax={\frac {i}{2}}\ln \left({\frac {\Gamma (x+{\frac {i}{a}})}{\Gamma (x-{\frac {i}{a}})}}\right)+C}$

Antidifferences of special functions

${\displaystyle \sum _{x}\psi (x)=(x-1)\psi (x)-x+C}$

${\displaystyle \sum _{x}\Gamma (x)=(-1)^{x+1}\Gamma (x){\frac {\Gamma (1-x,-1)}{e}}+C}$

where ${\displaystyle \Gamma (s,x)}$ is the incomplete gamma function.

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

where ${\displaystyle (x)_{a}}$ is the falling factorial.

${\displaystyle \sum _{x}\operatorname {sexp} _{a}(x)=\ln _{a}{\frac {(\operatorname {sexp} _{a}(x))'}{(\ln a)^{x}}}+C}$

(see super-exponential function)

See also

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