Feature model

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

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

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:

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

Definitions

Laplace summation formula

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

where $c_k=\underset{0}{\overset{1}{\int }}\frac{\mathrm{\Gamma }(x+1)}{\mathrm{\Gamma }(x-k+1)}dx$ are the Bernoulli numbers of the second kind.

Newton's formula

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

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

Faulhaber's formula

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

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

Mueller's formula

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

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

Ramanujan's formula

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

Connection to the Ramanujan summation

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

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

or

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

In this case, where

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

then Ramanjuan's sum is defined as

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

or

$\sum _{x\ge 1}^{\mathrm{\Re }}f(x)=F(1)$

Summation by parts

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.

Indefinite summation by parts:

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

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

Definite summation by parts:

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

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 $\mathrm{\nabla }$ 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

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

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

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

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

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

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

$\sum _x{\log }_{b}x={\log }_b\mathrm{\Gamma }(x)+C$

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

Antidifferences of hyperbolic functions

$\sum _x\sinh ax=\frac{1}{2}\mathrm{csch}\left(\frac{a}{2}\right)\cosh (\frac{a}{2}-ax)+C$

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

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

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

Antidifferences of trigonometric functions

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

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

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

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

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

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

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

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

Antidifferences of inverse hyperbolic functions

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

Antidifferences of inverse trigonometric functions

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

Antidifferences of special functions

$\sum _x\psi (x)=(x-1)\psi (x)-x+C$

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

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

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

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

$\sum _x{\mathrm{sexp}}_a(x)={\ln }_a\frac{({\mathrm{sexp}}_a(x){)}^{\prime }}{(\ln a{)}^x}+C$

(see super-exponential function)

See also

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

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

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