indefinite sum

In discrete calculus the indefinite sum operator (also known as the antidifference operator), denoted by $\sum _x$ or $\Delta^{-1}$ ,{{citation

| last = Man | first = Yiu-Kwong

| doi = 10.1006/jsco.1993.1053

| issue = 4

| journal = Journal of Symbolic Computation

| mr = 1263873

| pages = 355–376

| title = On computing closed forms for indefinite summations

| volume = 16

| year = 1993}}{{citation

| last = Goldberg | first = Samuel

| mr = 94249

| page = 41

| quote = If $Y$ is a function whose first difference is the function $y$ , then $Y$ is called an indefinite sum of $y$ and denoted by $\Delta^{-1}y$

| publisher = Wiley, New York, and Chapman & Hall, London

| title = Introduction to difference equations, with illustrative examples from economics, psychology, and sociology

| url = https://books.google.com/books?id=QUzNwiVpWGAC&pg=PA41

| year = 1958| isbn = 978-0-486-65084-5

}}; reprinted by Dover Books, 1986 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, 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: $\Delta^{-1}=\frac1{e^D-1}$ .

Fundamental theorem of discrete calculus

Indefinite sums can be used to calculate definite sums with the formula:"Handbook of discrete and combinatorial mathematics", Kenneth H. Rosen, John G. Michaels, CRC Press, 1999, {{ISBN|0-8493-0149-1}}

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

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:{{citation

| last1 = Merlini | first1 = Donatella | author1-link = Donatella Merlini

| last2 = Sprugnoli | first2 = Renzo

| last3 = Verri | first3 = M. Cecilia

| doi = 10.1016/j.disc.2006.03.065

| issue = 16

| journal = Discrete Mathematics

| mr = 2251571

| pages = 1906–1920

| title = The Cauchy numbers

| volume = 306

| year = 2006}}

$\sum_x = \int{} + \frac{1}{2} - \frac{1}{12}\Delta + \frac{1}{24}\Delta^2 - \frac{19}{720}\Delta^3 + \frac{3}{160}\Delta^4 - \cdots$

In this formula, for instance, the term $\tfrac12$ represents an operator that divides the given function by two. The coefficients $+\tfrac12, -\tfrac1{12},$ etc., appearing in this formula are the Gregory coefficients, also called Laplace numbers. The coefficient in the term $\Delta^{n-1}$ is

$\frac{\mathcal{C}_n}{n!}=\int_0^1 \binom{x}{n}\,dx$

where the numerator $\mathcal{C}_n$ of the left hand side is called a Cauchy number of the first kind, although this name sometimes applies to the Gregory coefficients themselves.

=Newton's formula=

: $\sum_x f(x)=\sum_{k=1}^\infty \binom{x}k \Delta^{k-1} [f]\left (0\right)+C=\sum_{k=1}^{\infty}\frac{\Delta^{k-1}[f](0)}{k!}(x)_k+C$

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

=Faulhaber's formula=

: $\sum _x f(x)= \sum_{n=1}^{\infty} \frac{f^{(n-1)} (0)}{n!} B_n(x) + C \, ,$

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

=Mueller's formula=

If $\lim_{x\to{+\infty}}f(x)=0,$ then[http://www.math.tu-berlin.de/~mueller/HowToAdd.pdf Markus Müller. How to Add a Non-Integer Number of Terms, and How to Produce Unusual Infinite Summations] {{webarchive|url=https://web.archive.org/web/20110617053801/http://www.math.tu-berlin.de/~mueller/HowToAdd.pdf |date=2011-06-17 }} (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)

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

=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

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

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

Alternatively, Ramanujan's sum can be used:

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

or at 1

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

respectivelyBruce C. Berndt, [http://www.comms.scitech.susx.ac.uk/fft/math/RamanujanNotebooks1.pdf Ramanujan's Notebooks] {{webarchive|url=https://web.archive.org/web/20061012064851/http://www.comms.scitech.susx.ac.uk/fft/math/RamanujanNotebooks1.pdf |date=2006-10-12 }}, Ramanujan's Theory of Divergent Series, Chapter 6, Springer-Verlag (ed.), (1939), pp. 133–149.Éric Delabaere, [http://algo.inria.fr/seminars/sem01-02/delabaere2.pdf Ramanujan's Summation], Algorithms Seminar 2001–2002, F. Chyzak (ed.), INRIA, (2003), pp. 83–88.

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:

: $\sum_{k=1}^n f(k).$

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.[http://www.risc.uni-linz.ac.at/people/mkauers/publications/kauers05c.pdf Algorithms for Nonlinear Higher Order Difference Equations], Manuel Kauers It is the inverse of the 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=

:For positive integer exponents Faulhaber's formula can be used. Note that x in the result must be replaced with x-1 due to the offset caused by the indefinite sum being defined the inverse of the forward difference operator. For negative integer exponents,

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

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

:More generally,

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

:where $\zeta(s,a)$ is the Hurwitz zeta function and $\psi(z)$ is the Digamma function. By considering this for negative a (indefinite sum over reciprocal powers), and adding 1 to x, this becomes the Generalized harmonic number. For further information, refer to Balanced polygamma function and Hurwitz zeta function#Special cases and generalizations. Further generalization comes from use of the Lerch transcendent:

: $\sum_x \frac{z^{x}}{(x+a)^{s}} = - z^{x} \, \Phi(z, s, x + a ) + C$

:Which generalizes the Generalized harmonic number. Additionally, the partial derivative is given by

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

: $\sum _x B_a(x)=(x-1)B_a(x)-\frac{a}{a+1} B_{a+1}(x)+C$

=Antidifferences of exponential functions=

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

=Antidifferences of logarithmic functions=

: $\sum _x \log_b x = \log_b (x!) + C$

: $\sum _x \log_b ax = \log_b (x!a^{x}) + C$

=Antidifferences of hyperbolic functions=

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

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

: $\sum _x \tanh ax = \frac1a \psi _{e^a}\left(x-\frac{i \pi }{2 a}\right)+\frac1a \psi _{e^a}\left(x+\frac{i \pi }{2 a}\right)-x + C$

:where $\psi_q(x)$ is the q-digamma function.

=Antidifferences of trigonometric functions=

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

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

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

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

: $\sum_x \tan ax = i x-\frac1a \psi _{e^{2 i a}}\left(x-\frac{\pi }{2 a}\right) + C \,,\,\,a\ne \frac{n\pi}2$

:where $\psi_q(x)$ is the q-digamma function.

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

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

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

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

=Antidifferences of inverse hyperbolic functions=

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

=Antidifferences of inverse trigonometric functions=

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

=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)