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

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

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

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

If F(x) is a solution of this functional equation for a given f(x), then so is F(x)+C for any constant C. Therefore each indefinite sum actually represents a family of functions, differing by an additive constant.

Fundamental theorem of discrete calculus

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

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

Definitions

Laplace summation formula

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

where $c_{k}=\int _{0}^{1}{\frac {\Gamma (x+1)}{\Gamma (x-k+1)}}dx$ are the Bernoulli numbers of the second kind.

Newton's formula

$\sum _{x}f(x)=-\sum _{k=1}^{\infty }{\frac {\Delta ^{k-1}f(x)}{k!}}(-x)_{k}+C$

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

Faulhaber's formula

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

provided that the right-hand side of the equation converges.

Mueller's formula

If $\lim _{x\to {+\infty }}f(x)=0,$ then

$\sum _{x}f(x)=\sum _{n=0}^{\infty }\left(f(n)-f(n+x)\right)+C.$

Ramanujan's formula

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

Connection to the Ramanujan summation

Often the constant C in indefinite sum is fixed from the following equation:

$\int _{1}^{2}\sum _{x}f(x)dx=0$

or

$\int _{0}^{1}\sum _{x}f(x)dx=0$

In this case, where

$F(x)=\sum _{x}f(x)\,$

then Ramanjuan's sum is defined as

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

or

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

Summation by parts

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Indefinite summation by parts:

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

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

Definite summation by parts:

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

Period rule

If $T\,$ is a period of function $f(x)\,$ then

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

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. e.g.

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

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

$F(x+1)-F(x)=f(x+1)\,$ which is called the telescoping equation. It is inverse to backward difference $\nabla$ operator.

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

List of indefinite sums

This is a list of indefinite sums of various functions. Not every function has an indefinite sum that can be expressed in terms of elementary functions.

Antidifferences of rational functions

$\sum _{x}a=ax+C\,$

$\sum _{x}x={\frac {x^{2}}{2}}-{\frac {x}{2}}+C$

$\sum _{x}x^{a}={\frac {B_{a+1}(x)}{a+1}}+C,\,a\notin \mathbb {Z} ^{-}$

where $B_{a}(x)=-a\zeta (-a+1,x)\,$, the generalized to real order Bernoulli polynomials.

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

where $\psi ^{(n)}(x)$ is the polygamma function.

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

where $\psi (x)$ is the digamma function.

Antidifferences of exponential functions

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

Antidifferences of logarithmic functions

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

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

Antidifferences of hyperbolic functions

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

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

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

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

Antidifferences of trigonometric functions

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

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

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

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

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

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

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

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

Antidifferences of inverse hyperbolic functions

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

Antidifferences of inverse trigonometric functions

$\sum _{x}\arctan ax={\frac {i}{2}}\ln \left({\frac {(-1)^{x}\Gamma ({\frac {-i}{a}})\Gamma (x+{\frac {i}{a}})}{\Gamma ({\frac {i}{a}})\Gamma (x-{\frac {i}{a}})}}\right)+C$

Antidifferences of special functions

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

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

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

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

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

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

(see super-exponential function)

See also

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

References

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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