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| In [[mathematics]] the '''indefinite sum''' operator (also known as the '''antidifference''' operator), denoted by <math>\sum _x \,</math> or <math>\Delta^{-1} \,</math>,<ref>{{PlanetMath|urlname=IndefiniteSum|title=Indefinite Sum}}</ref><ref>[http://hostel6.ru/books/_Papers/Computer_algebra/Summation/Man.%20Closed%20forms%20for%20symbolic%20summation.%20JSC%201993%20(22s).pdf On Computing Closed Forms for Indefinite Summations. Yiu-Kwong Man. J. Symbolic Computation (1993), 16, 355-376]</ref><ref>"If ''Y'' is a function whose first difference is the function ''y'', then ''Y'' is called an indefinite sum of ''y'' and denoted Δ<sup>-1</sup>''y''" [http://books.google.co.uk/books?id=5rFOeE0zvY4C&pg=PA41&dq=%22indefinite+sum%22 ''Introduction to Difference Equations''], Samuel Goldberg</ref> is the [[linear operator]], inverse of the [[difference operator|forward difference operator]] <math>\Delta \,</math>. It relates to the [[difference operator|forward difference operator]] as the [[indefinite integral]] relates to the [[derivative]]. Thus
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| :<math>\Delta \sum_x f(x) = f(x) \, .</math>
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| More explicitly, if <math>\sum_x f(x) = F(x) \,</math>, then
| | By investing in a premium Word - Press theme, you're investing in the future of your website. It is thus, on these grounds that compel various web service provider companies to integrate the same in their packages too. Should you go with simple HTML or use a platform like Wordpress. Keep reading for some great Word - Press ideas you can start using today. You can customize the appearance with PSD to Word - Press conversion ''. <br><br>Generally, for my private income-making market websites, I will thoroughly research and discover the leading 10 most worthwhile niches to venture into. You will have to invest some money into tuning up your own blog but, if done wisely, your investment will pay off in the long run. It sorts the results of a search according to category, tags and comments. So, if you are looking for some option to build a giant e-commerce website, then e-shopping preferable CMS tools will be helpful for you. Akismet is really a sophisticated junk e-mail blocker and it's also very useful thinking about I recieve many junk e-mail comments day-to-day across my various web-sites. <br><br>You can down load it here at this link: and utilize your FTP software program to upload it to your Word - Press Plugin folder. If you loved this write-up and you would certainly like to obtain even more details pertaining to [http://www.ase.re/wordpress_dropbox_backup_7996251 backup plugin] kindly check out our own web site. But if you are not willing to choose cost to the detriment of quality, originality and higher returns, then go for a self-hosted wordpress blog and increase the presence of your business in this new digital age. I hope this short Plugin Dynamo Review will assist you to differentiate whether Plugin Dynamo is Scam or a Genuine. Enough automated blog posts plus a system keeps you and your clients happy. Socrates: (link to ) Originally developed for affiliate marketers, I've used this theme again and again to develop full-fledged web sites that include static pages, squeeze pages, and a blog. <br><br>The disadvantage is it requires a considerable amount of time to set every thing up. I didn't straight consider near it solon than one distance, I got the Popup Ascendancy plugin and it's up and lengthways, likely you make seen it today when you visited our blog, and I yet customize it to fit our Thesis Wound which gives it a rattling uncomparable visage and search than any different popup you know seen before on any added journal, I hump arrogated asset of one of it's quatern themes to make our own. A higher percentage of women are marrying at older ages,many are delaying childbearing until their careers are established, the divorce rate is high and many couples remarry and desire their own children. IVF ,fertility,infertility expert,surrogacy specialist in India at Rotundaivf. Look for experience: When you are searching for a Word - Press developer you should always look at their experience level. <br><br>More it extends numerous opportunities where your firm is at comfort and rest assured of no risks & errors. Mahatma Gandhi is known as one of the most prominent personalities and symbols of peace, non-violence and freedom. By the time you get the Gallery Word - Press Themes, the first thing that you should know is on how to install it. Page speed is an important factor in ranking, especially with Google. For your information, it is an open source web content management system. |
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| :<math>F(x+1) - F(x) = f(x) \, .</math>
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| 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.
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| ==Fundamental theorem of discrete calculus==
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| Indefinite sums can be used to calculate definite sums with the formula:<ref>"Handbook of discrete and combinatorial mathematics", Kenneth H. Rosen, John G. Michaels, CRC Press, 1999, ISBN 0-8493-0149-1</ref>
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|
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| :<math>\sum_{k=a}^b f(k)=\Delta^{-1}f(b+1)-\Delta^{-1}f(a)</math>
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| ==Definitions==
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| ===Laplace summation formula===
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| :<math>\sum _x f(x)=\int_0^x f(t) dt +\sum_{k=1}^\infty \frac{c_k\Delta^{k-1}f(x)}{k!} + C </math>
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| :where <math>c_k=\int_0^1 \frac{\Gamma(x+1)}{\Gamma(x-k+1)}dx</math> are the [[Bernoulli numbers of the second kind]].<ref>[http://mathworld.wolfram.com/BernoulliNumberoftheSecondKind.html Bernoulli numbers of the second kind on Mathworld]</ref>
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| ===Newton's formula===
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| :<math>\sum_x f(x)=-\sum_{k=1}^{\infty}\frac{\Delta^{k-1}f(x)}{k!}(-x)_k+C</math>
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| :where <math>(x)_k=\frac{\Gamma(x+1)}{\Gamma(x-k+1)}</math> is the [[falling factorial]].
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| ===Faulhaber's formula===
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| :<math>\sum _x f(x)= \sum_{n=1}^{\infty} \frac{f^{(n-1)} (0)}{n!} B_n(x) + C \, ,</math>
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| provided that the right-hand side of the equation converges.
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| ===Mueller's formula===
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| If <math>\lim_{x\to{+\infty}}f(x)=0,</math> then
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| :<math>\sum _x f(x)=\sum_{n=0}^\infty\left(f(n)-f(n+x)\right)+ C.</math> <ref>[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] (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)</ref>
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| ===Ramanujan's formula===
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| :<math>\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</math>
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| ==Connection to the Ramanujan summation==
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| Often the constant C in indefinite sum is fixed from the following equation:
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| : <math>\int_1^2 \sum _x f(x) dx=0 </math>
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| or
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| : <math>\int_0^1 \sum _x f(x) dx=0 </math>
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| In this case, where
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| : <math>F(x)=\sum _x f(x) \,</math>
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| then Ramanjuan's sum is defined as | |
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| : <math>\sum_{x \ge 1}^{\Re}f(x)=F(0)\,</math>
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| or
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| : <math>\sum_{x \ge 1}^{\Re}f(x)=F(1)\,</math><ref>Bruce C. Berndt, [http://www.comms.scitech.susx.ac.uk/fft/math/RamanujanNotebooks1.pdf Ramanujan's Notebooks], ''Ramanujan's Theory of Divergent Series'', Chapter 6, Springer-Verlag (ed.), (1939), pp. 133–149.</ref><ref>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.</ref>
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| ==Summation by parts==
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| {{main|Summation by parts}}
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| Indefinite summation by parts:
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| :<math>\sum_x f(x)\Delta g(x)=f(x)g(x)-\sum_x (g(x)+\Delta g(x)) \Delta f(x) \,</math>
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| :<math>\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) \,</math>
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| Definite summation by parts:
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| :<math>\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)</math>
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| ==Period rule==
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| If <math>T \,</math> is a period of function <math>f(x)\,</math> then
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| :<math>\sum _x f(Tx)=x f(Tx) + C\,</math> | |
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| ==Alternative usage==
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| 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.
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| :<math>\sum_{k=1}^n f(k)</math>
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| In this case a closed form expression F(k) for the sum is a solution of
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| :<math>F(x+1) - F(x) = f(x+1) \,</math> which is called the telescoping equation.<ref>[http://www.risc.uni-linz.ac.at/people/mkauers/publications/kauers05c.pdf Algorithms for Nonlinear Higher Order Difference Equations], Manuel Kauers</ref> It is inverse to [[backward difference]] <math>\nabla</math> operator.
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| It is related to the forward antidifference operator using the fundamental theorem of discrete calculus described earlier.
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| ==List of indefinite sums==
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| 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.
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| ===Antidifferences of rational functions===
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| :<math>\sum _x a = ax + C \,</math>
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| :<math>\sum _x x = \frac{x^2}{2}-\frac{x}{2} + C</math>
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| :<math>\sum _x x^a = \frac{B_{a+1}(x)}{a+1} + C,\,a\notin \mathbb{Z}^-</math> | |
| :where <math>B_a(x)=-a\zeta(-a+1,x)\,</math>, the generalized to real order [[Bernoulli polynomials]].
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| :<math>\sum _x x^a = \frac{(-1)^{a-1}\psi^{(-a-1)}(x)}{\Gamma(-a)}+ C,\,a\in\mathbb{Z}^-</math>
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| :where <math>\psi^{(n)}(x)</math> is the [[polygamma function]].
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| :<math>\sum _x \frac1x = \psi(1-x) + C </math>
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| :where <math>\psi(x)</math> is the [[digamma function]].
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| ===Antidifferences of exponential functions===
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| :<math>\sum _x a^x = \frac{a^x}{a-1} + C \,</math>
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| ===Antidifferences of logarithmic functions===
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| :<math>\sum _x \log_b x = \log_b \Gamma (x) + C \,</math>
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| :<math>\sum _x \log_b ax = \log_b (a^{x-1}\Gamma (x)) + C \,</math>
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| ===Antidifferences of hyperbolic functions===
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| :<math>\sum _x \sinh ax = \frac{1}{2} \operatorname{csch} \left(\frac{a}{2}\right) \cosh \left(\frac{a}{2} - a x\right) + C \,</math>
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| :<math>\sum _x \cosh ax = \frac{1}{2} \coth \left(\frac{a}{2}\right) \sinh ax -\frac{1}{2} \cosh ax + C \,</math>
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| :<math>\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</math>
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| :where <math>\psi_q(x)</math> is the [[q-analog|q-digamma]] function.
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| ===Antidifferences of trigonometric functions===
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| :<math>\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 </math>
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| :<math>\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</math>
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| :<math>\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</math>
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| :<math>\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</math>
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| :<math>\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</math>
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| :where <math>\psi_q(x)</math> is the [[q-analog|q-digamma]] function.
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| :<math>\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\,</math>
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| :<math>\sum_x \cot ax =-i x-\frac{i \psi _{e^{2 i a}}(x)}{a} + C \,,\,\,a\ne \frac{n\pi}2</math>
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| ===Antidifferences of inverse hyperbolic functions===
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| :<math>\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</math> | |
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| ===Antidifferences of inverse trigonometric functions===
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| :<math>\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</math>
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| ===Antidifferences of special functions===
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| :<math>\sum _x \psi(x)=(x-1) \psi(x)-x+C \,</math>
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| :<math>\sum _x \Gamma(x)=(-1)^{x+1}\Gamma(x)\frac{\Gamma(1-x,-1)}e+C</math>
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| :where <math>\Gamma(s,x)</math> is the [[incomplete gamma function]].
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| :<math>\sum _x (x)_a = \frac{(x)_{a+1}}{a+1}+C</math>
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| :where <math>(x)_a</math> is the [[falling factorial]].
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| :<math>\sum _x \operatorname{sexp}_a (x) = \ln_a \frac{(\operatorname{sexp}_a (x))'}{(\ln a)^x} + C \,</math>
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| :(see [[super-exponential function]])
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| ==See also==
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| *[[Indefinite product]]
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| *[[Time scale calculus]]
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| *[[List of derivatives and integrals in alternative calculi]]
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| * "Difference Equations: An Introduction with Applications", Walter G. Kelley, Allan C. Peterson, Academic Press, 2001, ISBN 0-12-403330-X
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| * [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]
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| * [http://arxiv.org/abs/math/0502109 Markus Mueller, Dierk Schleicher. Fractional Sums and Euler-like Identities]
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| * [http://www.springerlink.com/content/kj0jx24240756457/ S. P. Polyakov. Indefinite summation of rational functions with additional minimization of the summable part. Programmirovanie, 2008, Vol. 34, No. 2.]
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| * "Finite-Difference Equations And Simulations", Francis B. Hildebrand, Prenctice-Hall, 1968
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| {{DEFAULTSORT:Indefinite Sum}}
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| [[Category:Mathematical analysis]]
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| [[Category:Mathematical tables|Indefinite sums]]
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| [[Category:Finite differences]]
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| [[Category:Linear operators in calculus]]
| |
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