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

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

Laplace summation formula edit

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

where ${\displaystyle c_{k}=\int _{0}^{1}{\frac {\Gamma (x+1)}{\Gamma (x-k+1)}}dx}$  are the Cauchy numbers of the first kind, also known as the Bernoulli Numbers of the Second Kind.^{[4][citation needed]}

Newton's formula edit

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

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

Faulhaber's formula edit

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

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

Mueller's formula edit

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

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

Euler–Maclaurin formula edit

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

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]

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:

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

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

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

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

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

In this case a closed form expression F(k) 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 ${\displaystyle \nabla }$  operator. It is related to the forward antidifference operator using the fundamental theorem of discrete calculus described earlier.

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 edit

${\displaystyle \sum _{x}a=ax+C}$ 

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

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

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

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

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

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

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

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

Antidifferences of exponential functions edit

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

Particularly,

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

Antidifferences of logarithmic functions edit

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

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

Antidifferences of hyperbolic functions edit

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

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

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

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

Antidifferences of trigonometric functions edit

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

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

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

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

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

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 edit

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

Antidifferences of inverse trigonometric functions edit

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

Antidifferences of special functions edit

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

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

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)

• 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