# Indefinite sum

(Redirected from Antidifference)

In mathematics the indefinite sum operator (also known as the antidifference operator), denoted by $\sum _x \,$ or $\Delta^{-1} \,$,[1][2][3] 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(x) for any periodic function C(x) with period 1. Therefore each indefinite sum actually represents a family of functions. However the solution equal to its Newton series expansion is unique up to an additive constant C.

## Fundamental theorem of discrete calculus

Indefinite sums can be used to calculate definite sums with the formula:[4]

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

## 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.[5]

### 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[6]

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

### Euler–Maclaurin formula

$\sum _x f(x)= \int_0^x f(t) dt - \frac12 f(x)+\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}f^{(2k-1)}(x) + C$

## Choice of the constant term

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

Let

$F(x)=\sum _x f(x)+C$

Then the constant C is fixed from the condition

$\int_0^1 F(x) dx=0$

or

$\int_1^2 F(x) dx=0$

Alternatively, Ramanujan's sum can be used:

$\sum_{x \ge 1}^{\Re}f(x)=-f(0)-F(0)$

or at 1

$\sum_{x \ge 1}^{\Re}f(x)=-F(1)$

respectively[7][8]

## Summation by parts

Main article: Summation by parts

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 rules

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

$\sum _x f(Tx)=x f(Tx) + C\,$

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

$\sum _x f(Tx)=-\frac12 f(Tx) + C\,$

## 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. 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.[9] 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.
$\sum _x \frac1x = \psi(x) + C$
where $\psi(x)$ is the digamma function.

### Antidifferences of exponential functions

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

Particularly,

$\sum _x 2^x = 2^x + C \,$

### 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

$\sum _x \sinh ax = \frac{1}{2} \operatorname{csch} \left(\frac{a}{2}\right) \cosh \left(\frac{a}{2} - a x\right) + C \,$
$\sum _x \cosh ax = \frac{1}{2} \coth \left(\frac{a}{2}\right) \sinh ax -\frac{1}{2} \cosh ax + C \,$
$\sum _x \tanh ax = \frac1a \psi _{e^a}\left(x-\frac{i \pi }{2 a}\right)+\frac1a \psi _{e^a}\left(x+\frac{i \pi }{2 a}\right)-x + C$
where $\psi_q(x)$ is the q-digamma function.

### Antidifferences of trigonometric functions

$\sum _x \sin ax = -\frac{1}{2} \csc \left(\frac{a}{2}\right) \cos \left(\frac{a}{2}- a x \right) + C \,,\,\,a\ne n \pi$
$\sum _x \cos ax = \frac{1}{2} \cot \left(\frac{a}{2}\right) \sin ax -\frac{1}{2} \cos ax + C \,,\,\,a\ne n \pi$
$\sum _x \sin^2 ax = \frac{x}{2} + \frac{1}{4} \csc (a) \sin (a-2 a x) + C \, \,,\,\,a\ne \frac{n\pi}2$
$\sum _x \cos^2 ax = \frac{x}{2}-\frac{1}{4} \csc (a) \sin (a-2 a x) + C \,\,,\,\,a\ne \frac{n\pi}2$
$\sum_x \tan ax = i x-\frac1a \psi _{e^{2 i a}}\left(x-\frac{\pi }{2 a}\right) + C \,,\,\,a\ne \frac{n\pi}2$
where $\psi_q(x)$ is the q-digamma function.
$\sum_x \tan x=ix-\psi _{e^{2 i}}\left(x+\frac{\pi }{2}\right) + C = -\sum _{k=1}^{\infty } \left(\psi \left(k \pi -\frac{\pi }{2}+1-z\right)+\psi \left(k \pi -\frac{\pi }{2}+z\right)-\psi \left(k \pi -\frac{\pi }{2}+1\right)-\psi \left(k \pi -\frac{\pi }{2}\right)\right) + C\,$
$\sum_x \cot ax =-i x-\frac{i \psi _{e^{2 i a}}(x)}{a} + C \,,\,\,a\ne \frac{n\pi}2$

### Antidifferences of inverse hyperbolic functions

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

### Antidifferences of inverse trigonometric functions

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

### 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)

## References

1. ^
2. ^ On Computing Closed Forms for Indefinite Summations. Yiu-Kwong Man. J. Symbolic Computation (1993), 16, 355-376
3. ^ "If Y is a function whose first difference is the function y, then Y is called an indefinite sum of y and denoted Δ−1y" Introduction to Difference Equations, Samuel Goldberg
4. ^ "Handbook of discrete and combinatorial mathematics", Kenneth H. Rosen, John G. Michaels, CRC Press, 1999, ISBN 0-8493-0149-1
5. ^ Bernoulli numbers of the second kind on Mathworld
6. ^ Markus Müller. How to Add a Non-Integer Number of Terms, and How to Produce Unusual Infinite Summations (note that he uses a slightly alternative definition of fractional sum in his work, i.e. inverse to backwards difference, hence 1 as the lower limit in his formula)
7. ^ Bruce C. Berndt, Ramanujan's Notebooks, Ramanujan's Theory of Divergent Series, Chapter 6, Springer-Verlag (ed.), (1939), pp. 133–149.
8. ^ Éric Delabaere, Ramanujan's Summation, Algorithms Seminar 2001–2002, F. Chyzak (ed.), INRIA, (2003), pp. 83–88.
9. ^ Algorithms for Nonlinear Higher Order Difference Equations, Manuel Kauers