Talk:Indefinite sum

Title

I feel like the title Antidifference makes more sense then Indefinite sum TheTrueSauce (talk) 21:09, 17 April 2023 (UTC)

Agreed. But the term "indefinite sum" is used in Wolfram products.--Reciprocist (talk) 04:33, 13 December 2023 (UTC)

Higher-order antidifferences and the Nørlund–Rice integral

The current article only covers the first‑order antidifference. Higher‑order antidifferences are not yet mentioned as thoroughly as they are in Finite difference.

The Nørlund–Rice integral gives an integral representation of $\Delta ^{n}f(x)$. Its formal inverse is therefore an integral representation of the nth‑order antidifference and may be related to this article, but I am not aware of literature that presents it. Sure Beae (talk) 20:38, 12 February 2026 (UTC)

Jordan page 102 might be useful. Right after 102, on 104, citations viable for the List are present. 102 might also be useful for Newton series, if a generalization (there are many) can be chosen. Sure Beae (talk) 07:13, 13 February 2026 (UTC)

Actually, Nørlund seems to provide the logic for this at the start of page 139? If so, this should be an easy addition. Sure Beae (talk) 09:30, 18 February 2026 (UTC)

"Die Hauptlösungen im komplexen Gebiet" (the principal solutions in the complex domain) and "Mehrfache Summen" (multiple sums), along with specifically pages 138 to 139 (for specific logic), will be our likely citations. Sure Beae (talk) 20:56, 18 February 2026 (UTC)

https://arxiv.org/abs/2602.23025 Repeated principal indefinite summation by Thomas Lamby and Jean-Luc Marichal Sure Beae (talk) 18:44, 7 July 2026 (UTC)

Definitely solves our problem for the real analysis case, so it should be noted in the real analysis section. Multiple sums from Nørlund combined with Abel-Plana would solve our integer higher-order antidifferences coverage case. Directly relevant to the non integer higher-order case is fractional calculus. Read the following papers if interested

https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70351

A fractional residue theorem and its applications in calculating real integrals

https://link.springer.com/content/pdf/10.1007/s00013-021-01654-5.pdf

Fractional differential relations for the Lerch zeta function

Both share coauthor Arran Fernandez.

Again, main blocker is synthesis level that is acceptable. Nørlund+Abel-Plana should be safe. Non integer coverage is harder to justify. Nørlund–Rice integrals in specific cases might actually be doing the mean operator, not an inverse finite difference one, or something else. Need to be especially careful there. Unquestionably relevant, given Rice's work was related to similar work from Jagerman. Note they historically both worked with/for Bell Labs (Jagerman more of a contractor). Sure Beae (talk) 22:47, 7 July 2026 (UTC)

https://www.mdpi.com/2227-7390/6/1/6

Weyl and Marchaud Derivatives: A Forgotten History is also related. Perhaps suitable for Further reading. Also not suitable for in article coverage, but good to keep in mind. Sure Beae (talk) 17:57, 15 July 2026 (UTC)

https://www.researchgate.net/publication/270276905_A_Unified_Approach_to_Fractional_Derivatives A unified approach to fractional derivatives - Manuel D. Ortigueira and Juan J. Trujillo

https://arxiv.org/abs/1503.06211 On The Indefinite Sum In Fractional Calculus - James Nixon

Oh, I forgot to state the logic:

If we are to figure out/find papers on non integer orders for higher-order antidifferences it's probably going to be from a paper relating to fractional calculus, and do it accidentally. So the authors should be watched for papers relating to the calculus of finite differences, as it might be the indefinite sum operator disguised under some different wording.

That's why fractional calculus papers that produce relevant results to series expansions or the rightmost half-plane solution are to be posted under this topic. Sure Beae (talk) 13:25, 21 July 2026 (UTC)

Remainder terms

The Newton series and Laplace summation sections currently lack the remainder term for truncating (replacing $\infty$ in the upper limit with a finite term) the given expressions. This makes said sections somewhat incomplete. Milne pages 5 through 20 and 57 through 62 can be used for the Newton series along with other resources.

After both are addressed, reverting the supersection from "Expansions and Definitions" back to "Definitions" should be considered. Sure Beae (talk) 20:46, 12 February 2026 (UTC)

We should expand the Newton series section using Nørlund pages 222-233 most likely. Pages 229-230 are very useful:

$F(x)=\sum _{s=0}^{n-1}a_{s}{\binom {x-1}{s}}+R_{n}(x)$

$a_{s}={\frac {s!}{2\pi i}}\int _{C}{\frac {F(z)}{(z-1)(z-2)\cdots (z-s-1)}}\,dz$

$R_{n}(x)={\frac {1}{2\pi i}}\int _{C}{\frac {F(z)}{(z-1)(z-2)\cdots (z-n)}}{\frac {dz}{z-x}}$

$R_{n}={\frac {1}{2\pi i}}{\frac {(-1)^{n}\,\Gamma (n+1-x)}{\Gamma (1-x)}}\int {\frac {\Gamma (\alpha +z-n)}{\Gamma (\alpha +z)}}{\frac {F(\alpha +z)}{z-(x-\alpha )}}\,dz$

$R_{n}={\frac {1}{2\pi i}}{\frac {\Gamma (n+1-x)}{\Gamma (1-x)}}\int {\frac {\Gamma (1-z-\alpha )}{\Gamma (n+1-z-\alpha )}}{\frac {F(\alpha +z)}{z-(x-\alpha )}}\,dz$

Even if we do not include it, referencing/citing where to look could be useful to people reading the article as a reference. Sure Beae (talk) 17:23, 6 March 2026 (UTC)

Sister article needed for alternating operator

We need to cover the indefinite alternating sum/inverse of the other operator Nørlund defined (alternating averaging) in a separate article for full coverage on Wikipedia.

The classical Bernoulli numbers are defined as the partial derivative with respect to $x$ of the inverse forward difference of $x^{s}$ evaluated at the origin (for whoever inherits this project in the future, see https://en.wikipedia.org/wiki/User:Sure_Beae/Math_notes for full context as the literature uses generating functions which expand around this point; it will be pedagogically useful for you to maintain this article, though it should not be explicitly stated within the article unless you find sourcing and it's not too textbook like). Bernoulli numbers extend naturally under our Abel-Plana formula, Euler numbers extend naturally via averaging operator and alternating Abel-Plana. Sure Beae (talk) 00:15, 27 May 2026 (UTC)

https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/generalized-averaging-operator/1E3B4871AF9889D35FBE7F2BCE844D1F - A Generalized Averaging Operator:

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/1E3B4871AF9889D35FBE7F2BCE844D1F/S0008414X00036968a.pdf/a-generalized-averaging-operator.pdf - PDF version. Sure Beae (talk) 00:31, 17 September 2026 (UTC)

Resources

The paper https://link.springer.com/article/10.1007/s13398-025-01730-3 is a decent resource to cite in the article. Its references are also decent resources.

Sourcing can be even further improved with Difference Equations with Applications to Queues by David L. Jagerman and through the paper itself and its other sources. Sure Beae (talk) 16:40, 11 June 2026 (UTC)

Jordan, Charles - https://archive.org/details/calculusoffinite0000unse/mode/2up

Steffensen, Johan Frederik - https://archive.org/details/interpolation0000unse/mode/2up

L.M. Milne-Thomson - https://archive.org/details/in.ernet.dli.2015.211803/mode/2up

Boole, George - https://archive.org/details/in.ernet.dli.2015.206061/mode/2up

Niels Erik Nørlund - https://archive.org/details/vorlesungenuberd0013niel/mode/2up Sure Beae (talk) 01:46, 28 June 2026 (UTC)

https://link.springer.com/book/10.1007/978-1-4613-0071-7

https://link-springer-com.wikipedialibrary.idm.oclc.org/book/10.1007/978-1-4613-0071-7 Sure Beae (talk) 01:56, 28 June 2026 (UTC)

https://www.taylorfrancis.com/books/mono/10.1201/9780203909737/difference-equations-applications-queues-david-jagerman has a preview PDF button that is useful.

https://books.google.com/books?id=XGIeBWGXlZgC&printsec=frontcover&source=gbs_ViewAPI#v=onepage&q&f=false is similar.

Difference Equations with Applications to Queues by David L. Jagerman seems to exist and is public facing on lib [dot] ysu [dot] am. This does not seem to be a legally distributed copy. You shouldn't link it here.

You may be able to borrow it from a university or other library:

https://search.worldcat.org/title/49569035 Sure Beae (talk) 02:06, 28 June 2026 (UTC)

Possible uses for the Multiplication formula for the Nörlund sum operator and their applications paper itself include citing series expansions (maybe) and also relating the subject to number theory, p-adic arithmetic, the Volkenborn integral, etc. in the Expansions and Applications sections. Sure Beae (talk) 19:31, 11 July 2026 (UTC)

https://archive.org/details/cu31924059412902/page/n9/mode/2up A treatise on differential equations, and on the calculus of finite differences - Hymers, J. (John) Sure Beae (talk) 20:52, 13 July 2026 (UTC)

Further reading: https://archive.org/search?query=+Abelsche+Funktionen+und+algebraische+Geometrie+&tab=all - Abelsche Funktionen Und Algebraische Geometrie by Fabio Conforto.

It's on algebraic geometry but it shows how that subject relates to Nørlund. Perhaps useful for Applications. Would be a stretch. Additional sources would be welcome to be able to add a highly specific Applications subsection for it. Sure Beae (talk) 10:14, 17 July 2026 (UTC)

Divergent Series - G.H. Hardy https://archive.org/details/dli.ernet.285939 potential for the Euler–Maclaurin formula section and some simple transforms, along with simple recurrences like Hurwitz zeta (Jagerman provides the most general version for Hurwitz zeta already, for arbitrary lower bound, though we use Candelpergher mainly right now). Sure Beae (talk) 05:19, 24 July 2026 (UTC)

https://arxiv.org/pdf/2402.03372 On a Simple Continuation for Partial Sums, Kamal Saleh

https://arxiv.org/pdf/1209.5739 Summability Calculus, Ibrahim M. Alabdulmohsin Sure Beae (talk) 17:43, 14 August 2026 (UTC)

I have reviewed the "On a Simple Continuation for Partial Sums" paper and its predecessor "Summability Calculus". Both are unsuitable for citation as mathematical references in this article. Heuristic arguments and faulty proofs. These aren't suitable for citations, probably not even for the "Further reading" section. Some results are interesting, but after reading through them, these are low quality/mostly uncitable. We must rely on actual real or complex analysis sources, ideally secondary sources or primary sources published to journals.

I found this paper via the video "Sum of a Non-Integer Number of Terms" (9naIxxeNdMk) from the Zundamon's Theorem channel, which listed the paper in its description. After reading the paper and its predecessor, it appears this channel makes no effort distinguish between rigorous papers and garbage (heuristic/sketchy) preprints. I recommend caution evaluating any further sources that channel cites if it continues to cover this topic. Sure Beae (talk) 16:30, 9 September 2026 (UTC)

We can perhaps use https://www.jstor.org/stable/1967976?seq=1 which is A Generalization of the Calculus of Finite Differences to Include the Differential Calculus by J. P. Ballantine (logic/structure/algebra side) plus Nørlund's Differenzenrechnung lectures chapter 3 section 6 where the limit of $\omega$ is taken to zero (analysis/brute force side) to state and show that infinitesimal calculus is a special case of the calculus of finite differences. Need to be explicit about the differences in their notations and the article's. Nørlund's notation is ignored as explained in archive 1 under Article scope, as $\nabla$ is no longer used as the mean operator. The article should probably have a warning of some sort about that, too. Nørlund also covers how branch cuts and such emerge. I'd love to avoid over-relying on just Nørlund, but him and his source material (which I shouldn't use, need secondary sources ideally) are far beyond the modern research; his book seems forgotten/unreferenced/not read or used in the current literature. Sure Beae (talk) 04:31, 28 August 2026 (UTC)

https://www.cambridge.org/core/journals/journal-of-the-australian-mathematical-society/article/summability-of-formal-solutions-of-functional-equations/B3B65E3608DC1E14A4AE87E3FD7DAF5D - The summability of formal solutions of functional equations:

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/B3B65E3608DC1E14A4AE87E3FD7DAF5D/S1446788700005516a.pdf/the-summability-of-formal-solutions-of-functional-equations.pdf - PDF version.

Very usable, especially for citing the principal solution terminology.

https://projecteuclid.org/journals/duke-mathematical-journal/volume-17/issue-1/A-note-on-the-summability-of-formal-solutions-of-linear/10.1215/S0012-7094-50-01706-6.short Bellman, R. cited there is also notable. Sure Beae (talk) 00:35, 17 September 2026 (UTC)

Higher-order antidifferences and period rules

See:

https://en.wikipedia.org/wiki/Talk:Indefinite_sum/Archive_1#Period_rules

https://en.wikipedia.org/wiki/Talk:Indefinite_sum/Archive_1#Period_rules_2

The content:

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

$\sum _{x}f(Tx)=xf(Tx)+C.$

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

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

Listed at https://en.wikipedia.org/wiki/List_of_indefinite_sums#Period_rules currently by Reciprocist without citation, which I myself or someone else needs to find and add. I do not know a source for this material. I might have seen it before as a Kelley and Peterson's Difference Equations as a practice problem. Reciprocist seems to desire it to be covered in the main article if possible.

The article cannot be a textbook per Wikipedia policy. However, if a higher-order antidifferences section is added, it might be possible to find context for it in a collapse block example of what the higher order antidifferences are unique up to. Consider for example $\Delta ^{2}x\sin \left(2\pi x\right)=0$. So the term for higher order antidifferences is not always simply 1-periodic. Other than that, to my knowledge, it resonates with the kernel of the difference operator, and is not the result of any of the article's Uniqueness subsections.

If anyone has potential relevant sources, I would greatly appreciate it. Sure Beae (talk) 18:25, 17 June 2026 (UTC)

Resonance types

There are multiple different types of resonance at $2\pi n$. One such type is $f\left(x\right)={\frac {\sin \left(2\pi x\right)}{\pi x}}$, as you can see here:

https://www.desmos.com/calculator/52gkdsfplp?backgroundColor=bbb&textColor=235656&invertedColors

Please change $h$ in this graph actively. You can also do this via https://codeberg.org/AzulBeae/norlundcalc or by hand if you prefer. This type of resonance causes there to be no unique solution up to a constant, though there are solutions of the same type.

This is different from the type of resonance that causes $\nabla ^{-1}\sin \left(ax\right)\delta x$ to diverge. Examples:

$\sum _{k=1}^{x}\sin \left(ak\right)$ extends to:

${\frac {\sin \left({\frac {a}{2}}\left(x+1\right)\right)\sin \left({\frac {a}{2}}x\right)}{\sin \left({\frac {a}{2}}\right)}}$

And $\sum _{k=1}^{x}\cos \left(ak\right)$ to:

${\frac {\cos \left({\frac {a}{2}}\left(x+1\right)\right)\sin \left({\frac {a}{2}}x\right)}{\sin \left({\frac {a}{2}}\right)}}$

https://www.desmos.com/calculator/13mzbq3x1i

Please change $a$ to around $6.283185...$ and to values around it.

As you can see, there are different flavours.

The operator we are inverting is

$\Delta ^{-1}={\frac {1}{e^{D}-1}},$

whose denominator vanishes when the derivative $D$ acts on a mode of frequency $2\pi n$, because $e^{2\pi in}-1=0$. What matters is whether the function $f$ actually contains that frequency as a pure tone, or merely as part of a continuous spread, or something else.

It would be very good to come up with sources explaining the different types of failure modes, if citations exist, especially in English, I would be very grateful for you leading me to them (I can be the one to actually add them if so desired). Sure Beae (talk) 06:52, 20 June 2026 (UTC)

As part of this issue, it might also be good to add the simple $\nabla ^{-1}\sin \left(ax\right)\delta x$ case as an example in the article in a collapsible block. Sure Beae (talk) 07:03, 20 June 2026 (UTC)

For anyone looking to rigorously formalize the distinction between the resonance types, this paper is the missing theoretical link:

https://arxiv.org/abs/1208.6079 - A unified approach to the integrals of Mellin–Barnes–Hecke type by Gopala Krishna Srinivasan.

It treats the pull-back of distributions and how common special functions can be derived through it, which aligns well with the theory of the calculus of finite differences presented in the article (in that you could use the theory presented as foundation for a paper that treats the general case with distribution theory without having to reach for the Abel–Plana formula).

It is quite dense in its current form and applying it directly to the antidifference resonance cases requires a significant synthesis (no mention of antidifferences or the calculus of finite differences directly in the paper, though it treats things that are trivially related like special functions) that falls outside the scope of this article (it would border on original research on my part, and, well, just be too long). However, it would make an excellent foundation for a dedicated research paper. I am fully open to anyone taking that idea and running with it without the need for credit; please just leave a note here so I can follow along and help integrate the findings into the article if it's ever written. Sure Beae (talk) 11:53, 28 June 2026 (UTC)

Find sources to link out to more special functions

Double factorial and it Double_factorial#Generalizations are quite literally step size generalizations applied to the indefinite product of $x$. In fact, unless it's related to Jean Écalle's work and resurgence theory, it is most typically a byproduct of Nørlund or a combination of higher order indefinite sums that you can reduce by rewriting with the shift operator. This includes Hurwitz and Riemann zeta. See https://en.wikipedia.org/wiki/User:Sure_Beae/Math_notes for derivation of Riemann and Hurwitz zeta functions (it's written in terms of a generalised Bernoulli number function and generalised shifted Bernoulli polynomials function, not directly). I would really appreciate citations for such things so the article can be more complete.

Double factorial: https://www.desmos.com/calculator/dkvbvkvkwb

$2^{{\frac {x}{2}}+{\frac {1}{4}}\left(1-\cos \left(\pi x\right)\right)}\pi ^{{\frac {1}{4}}\left(\cos \left(\pi x\right)-1\right)}\Gamma \left({\frac {x+2}{2}}\right)$.

Higher order composed linear difference equation: https://www.desmos.com/calculator/axz7gdtdvz

$F\left(x+2\right)+F\left(x+1\right)-F\left(x\right)-F\left(x-1\right)=x$

$F\left(x\right)={\frac {x^{2}}{8}}-{\frac {x}{8}}-{\frac {1}{16}}$ Sure Beae (talk) 23:16, 7 July 2026 (UTC)

$\ln \left(2^{{\frac {x}{2}}+{\frac {1}{4}}\left(1-\cos \left(\pi x\right)\right)}\pi ^{{\frac {1}{4}}\left(\cos \left(\pi x\right)-1\right)}\Gamma \left({\frac {x+2}{2}}\right)\right)-\ln \left(2^{{\frac {x-2}{2}}+{\frac {1}{4}}\left(1-\cos \left(\pi \left(x-2\right)\right)\right)}\pi ^{{\frac {1}{4}}\left(\cos \left(\pi \left(x-2\right)\right)-1\right)}\Gamma \left({\frac {x}{2}}\right)\right)=\ln \left(x\right)$ Sure Beae (talk) 15:12, 8 July 2026 (UTC)

https://viterbi-web.usc.edu/~adamchik/articles/polyg.pdf PolyGamma Functions of Negative Order - VICTOR S. ADAMCHIK

Direct hit for polygamma and for Barnes G. Sure Beae (talk) 10:52, 24 July 2026 (UTC)

Nørlund covers the higher order composed linear difference equations already of course but not to preferred depth for in article coverage. Sure Beae (talk) 02:25, 12 July 2026 (UTC)

Series expansions

Our coverage of linearity of the operator (in the summation by parts on $f$ sense) and series expansions is decent but not complete. I would really appreciate explicit sources/examples for Taylor or Laurent series expansions (to add example collapse blocks for it), though these were not explicitly calculated in the old sources they did provide the simplest series expansions like Maclaurin, they certainly justified it fully theory wise.

Examples of the kind of series expansion I am looking for here: https://github.com/surebeae/norlundcalc Sure Beae (talk) 23:30, 7 July 2026 (UTC)

https://link.springer.com/article/10.1007/s13398-025-01730-3 states a partial answer. I need something like $\left(-1\right)^{\left(a-1\right)}\left(\zeta \left(-a,1\right)-\zeta \left(-a,-x\right)\right)$, whatever the left half-plane solution is. The current citation to Vorlesungen über Differenzenrechnung pages 99 to 109 are decent but don't cover the full story. If I or someone else is to add this to the article, perhaps as its own section, they need to explicitly state the radii (or ovals?) of convergence on $F$. Sure Beae (talk) 02:54, 12 July 2026 (UTC)

$G(x)=(-1)^{a-1}\left(\zeta (-a,1)-\zeta (-a,-x)\right)$

$G(x-1)=(-1)^{a-1}\left(\zeta (-a,1)-\zeta (-a,-x+1)\right)$

$G(x)-G(x-1)=(-1)^{a-1}\left[\zeta (-a,-x+1)-\zeta (-a,-x)\right]$

Using the Hurwitz zeta recurrence: $\zeta (s,t+1)-\zeta (s,t)=-t^{-s},$

let $s=-a,\quad t=-x,$

$\zeta (-a,-x+1)-\zeta (-a,-x)=-(-x)^{-(-a)}=-(-x)^{a}$

$G(x)-G(x-1)=(-1)^{a-1}\cdot \left(-(-x)^{a}\right)$

$G(x)-G(x-1)=(-1)^{a-1}\cdot \left(-(-1)^{a}x^{a}\right)$

$G(x)-G(x-1)=(-1)^{a-1}\cdot (-1)^{a+1}x^{a}$

$G(x)-G(x-1)=(-1)^{2a}x^{a}$

$G(x)-G(x-1)=x^{a}$

It does indeed look like the left half-plane principal solution to me per the symmetry rules, but I need it cited in a source for the article as it's not as 'easy' or direct as the other examples (would likely not be passable as routine calculation, like the current content is), even if it's not shown in the series expansion context. Preferably also the shift by $-1/2$ of the real part of the origin of a Hurwitz-Taylor/Laurent series expansion obtains on its radius of convergence due to the symmetry the Bernoulli polynomials obey. Article would be much better for it, as it would allow the reader to expand from a point instead of having to find a step length clean maximal vertical strip to run Abel-Plana, or having to just use telescoping. Sure Beae (talk) 17:57, 12 July 2026 (UTC)

This does $G(-1)=0$ because if I tried $G(0)=0$ it'd work for specific cases like simple $x^{1}$ which agrees with the right half plane solution. But it'd fail to exist if I chose $G(0)=0$ and did $1/x$ because of the pole at the origin. The difference between the two half plane solutions is likely the periodic zeta function or closely related to it (take that remark with a grain of salt, going on vibes and wolfram series expansions alone).

Right (inverse backward $1/x$):

${\frac {\Pi '\left(x\right)}{\Pi \left(x\right)}}-{\frac {\Pi '\left(0\right)}{\Pi \left(0\right)}}$

Left (inverse backward $1/x$):

${\frac {\Pi '\left(-1-x\right)}{\Pi \left(-1-x\right)}}-{\frac {\Pi '\left(0\right)}{\Pi \left(0\right)}}$

Where capital Pi is the Gauss Pi function. Notice the pole for the left one.

As for the periodic zeta function observation:

Do ((-1)^(a-1))(HurwitzZeta(-a,1)-HurwitzZeta(-a,-x))-(HurwitzZeta(-a,1)-HurwitzZeta(-a,x+1)), a=-2 in WolframAlpha web to replicate.

Changing $a$ I was getting

1/3 π^2 (3 csc^2(π x) - 1)

π^3 cot(π x) csc^2(π x)

-1/6 π^4 (-2 csc^4(π x) - 4 cot^2(π x) csc^2(π x)) - π^4/45

-1/24 π^5 (-8 cot^3(π x) csc^2(π x) - 16 cot(π x) csc^4(π x))

-1/120 π^6 (-16 csc^6(π x) - 16 cot^4(π x) csc^2(π x) - 88 cot^2(π x) csc^4(π x)) - (2 π^6)/945

...

So on, so perhaps I could find papers that cover something similar to the difference of the generalized harmonic number (Hurwitz zeta form) and periodic zeta function to hopefully find a citation for at least the left half-plane solution. Sure Beae (talk) 18:11, 12 July 2026 (UTC)

Perhaps this can go in by citing page 21 which has $B_{\nu }(1-x)=(-1)^{\nu }B_{\nu }(x)$. Lehmer 1988 is also cited for this on Wolfram MathWorld, but that'd need to be checked and a page number found. Aside from that, Nörlund1924p19 (for the definition on page 18, which is in that citation, name was made before it was edited to include 18 for the defining recurrence), NörlundErgänzungssatz (for symmetry rules that let you get the other solution), and NörlundSingularity (which generalizes this to meromorphic functions) should be good enough to add it to Examples. Need to subtract 1 from each x to turn it into the inverse forward difference with the $\sum _{x}$ notation, but this part is doable. Doesn't solve the overall series citation issue, but it's a good start. Only new citation would be page 21. Sure Beae (talk) 10:33, 13 July 2026 (UTC)

Should also probably note somewhere in the article how they can use $F_{R}$ (right half-plane solution for $x^{a}$) to get $F_{L}$ (left half-plane solution) (literally just plugging in to the symmetry rules, the other $F$ has to satisfy the same difference equation and be of minimal exponential type, just for a different pole/branch cut set, so it obtains the other solution automatically). Sure Beae (talk) 02:59, 12 July 2026 (UTC)

Or the alternating identity for Bernoulli polynomials in this specific case. Sure Beae (talk) 03:00, 12 July 2026 (UTC)

Failed GA review fixes

I went through and systematically addressed verifiability (more than what was specifically mentioned) and grammar thoroughly on the 12th, later Headbomb made some tweaks to the citations. I renominated the article when I was finished the same day. We are currently waiting on a new GA review. The current topic can likely be ignored, there's nothing more to glean as far as I can tell (reread it, but cannot archive it as completed, so I will simply leave this note).

Any further feedback is welcome! Please respond with it to this topic, and I will get to it when I can. Sure Beae (talk) 23:10, 18 July 2026 (UTC)

Well, reading it over again, I/we can maybe make more lead changes, like a simple example or two, and attempt to reword and simplify. Past the basic examples, the actual content of the article needs to be advanced as real/complex analysis is required to fix the operator up to a constant rather than a span periodic component, but more effort could be made to make it relate more immediately to the reader and generally be more interesting. We could perhaps also add a warning/note stating that the article requires complex analysis and understanding Fourier series as a prerequisite? People have seen "Indefinite sum" and gotten confused on what the literature says that means vs their expectations of "sums":

https://en.wikipedia.org/wiki/Talk:Indefinite_sum/Archive_1#Article_scope Sure Beae (talk) 23:24, 18 July 2026 (UTC)

Span integral

Nørlund presents a few properties yet to be covered:

The generalized Ergänzungssatz needs better coverage (ideally, to show you can get other principal solutions via symmetries sometimes, in the case where $f$ is meromorphic, and explain the more general symmetry briefly mentioned but poorly explained in Symmetry of the principal solution [my bad, it's my own addition]).

Higher-order antidifferences (multiple summation/repeated summation).

Most importantly, the span integral: If we define $F$ such that it solves the inverse forward difference, we have, for example:

$F\left(x\right)={\frac {x\left(x-1\right)}{2}}$

$\int _{x}^{x+1}F\left(z\right)dz-\int _{0}^{1}F\left(z\right)dz=\int _{0}^{x}zdz$

The span integral is, generally speaking, the average value of the principal indefinite sum taken over exactly one interval of its span/step size:

${\frac {1}{\omega }}\int _{x}^{x+\omega }F(z)\,dz=\int _{a}^{x}\varphi (t)\,dt$

With a that addition, the article should be pretty complete in its coverage of Nørlund. The others would be nice to have, but this is a strong property that eventually needs full coverage. Sure Beae (talk) 02:26, 26 July 2026 (UTC)

I was thinking of putting it right below the Symmetry section, but I actually think it's best right after the Integral mean condition as its own full section, perhaps somewhat mentioned/referenced there in Integral mean to tie the two concepts in nicely, with how closely related they are. So putting it after Choice of constant term might make the most sense. Sure Beae (talk) 03:52, 26 July 2026 (UTC)

External links/norlundcalc

I will be adding https://github.com/surebeae/norlundcalc (implements the Nørlund principal solution) to the external links just like the desmos version I added there some time ago, because WolframAlpha, Maple, and others don't support non hypergeometric terms/principal solutions of other components/are more specialized. This is my own project, so it would be reasonable to be cautious about this link. As such, if anyone thinks it does not belong there, they can remove it. So far, it only implements the Abel–Plana formula with recurrence. In the future, the method described here:

https://math.stackexchange.com/questions/5143622/sources-on-hurwitz-power-series-expansions-for-indefinite-sums-antidifferences

will likely also be added. That MSE question is directly relevant to this article and its sourcing, as in it I ask for citations so I can cite the use of term-wise antidifferencing using Hurwitz zetas for the antidifference of the monomials in a Taylor or Laurent series expansion. Please see the above Series expansions topic. Sure Beae (talk) 10:17, 26 July 2026 (UTC)

Done.

Same for the desmos link. If anyone thinks that needs to be removed, please go ahead and remove it.

I will maybe also upload a copy paste of every single line of desmos graph's 'code' to a .txt file so it can be fully replicated in the future, upload that to a git repo, and archive the .txt code page to the internet archive, then post the links here eventually, so if the specific graph is ever lost it can be replicated and placed back into the external links easily (the logic written in desmos is rather simple, it could be rewritten into GeoGebra or another graphing calculator with minor tweaks most likely). Sure Beae (talk) 10:32, 26 July 2026 (UTC)

https://raw.githubusercontent.com/surebeae/destextforindefsum/refs/heads/main/graph.txt

Also, to address the addition of these links, because the general conflict of interest and external link policies suggest they might be violations:

Without these graphing tools, the article is quite abstract and difficult for mathematicians and students/general math enthusiasts (from emails where I asked for article feedback) to grasp. The Nørlund principal solution requires complex analysis, and standard commercial software (like Maple/Mathematica) does not implement this feature in a generic way for arbitrary meromorphic functions. I am addressing:

https://en.wikipedia.org/wiki/Talk:Indefinite_sum/Archive_1#Calculation_of_arbitrary_meromorphic_functions_antidifferences

The existing solutions are CAS, and do things symbolically with Karr/Gosper/etc. They are limited to hypergeometric terms (ratio of successive terms are rational functions), and are not representative of Nørlund's theory in full.

I add them under the spirit of the law and Ignore all rules policy. The external links policy is to prevent spam/commercial links. None of these are; these are routine calculations that make it easier for the reader to get direct feedback on what the operator actually is by graphing it. The code is open source, I don't gain anything from it (Wikipedia links don't provide search engine optimization). They were made with the intent to improve the article. If someone has a concern and wants to remove them, that is fair per the rules. Sure Beae (talk) 12:35, 26 July 2026 (UTC)

Horizontally thin strip confusion

I've had someone email me confused, likely from the 'Proof sketch for the simplest case (exponential type < 2π)' part of the Complex analysis (exponential type) section, thinking that a horizontally thin strip might not support a principal solution. This is simply not true, Nørlund's lectures don't prescribe a minimum horizontal width between pole real parts.

Example for $f(x)={\frac {1}{x}}+{\frac {1}{x-{\frac {1}{2}}}}$:

https://www.desmos.com/calculator/tfvcm0dkkk

$\Gamma \left(x\right)=\left(x-1\right)!$

$\psi \left(x\right)={\frac {\Gamma '\left(x\right)}{\Gamma \left(x\right)}}$

Define the right half-plane solution: $G\left(x\right)=\psi \left(x+1\right)-\psi \left(1\right)+\psi \left(x+{\frac {1}{2}}\right)-\psi \left({\frac {1}{2}}\right)$

Define the left half-plane solution via Ergänzungssatz: $F\left(x\right)=G\left(-x-{\frac {1}{2}}\right)$

Mean the two: $H\left(x\right)={\frac {G\left(x\right)+F\left(x\right)}{2}}$

Final formula for the inner solution between the real parts $0$ and $1/2$: $H\left(x\right)=\psi \left(x+1\right)+\psi \left(x+{\frac {1}{2}}\right)-\psi \left(1\right)-\psi \left({\frac {1}{2}}\right)+\pi \cot \left(2\pi x\right)$.

All three principal solutions satisfy the inverse backward difference for the given $f$.

So, no, the strip doesn't have a lower bound for how thin it can be per Nørlund's theory. He never states one. The article should state its assumptions for the proof sketch in or after it more clearly. I added the sketch in originally when the article was starved for rigor and simple explanations of the concepts. Now that we cite Nørlund explicitly, it might be better to eventually remove it? I somewhat believe it still holds pedagogical value in the relatively (conceptually) difficult article, so I personally vote to keep it. I'd like to hear feedback, does anyone think we should maybe remove it?

If not, how should the clarifying statement be worded/cited? I personally will cite to Nørlund's core theorems if no ideas are brought up, then archive this topic as completed once I get to it. Sure Beae (talk) 05:10, 27 July 2026 (UTC)

Oops! This actually has too high of local 'exponential type' (imaginary growth rate) in the strip. So the averaging method is incorrect. The actual inner solution for the open interval $(0,1/2)$ is $\psi \left(x+1\right)+\psi \left({\frac {1}{2}}-x\right)-\psi \left(1\right)-\psi \left({\frac {1}{2}}\right)$. Sure Beae (talk) 15:59, 6 September 2026 (UTC)

The correction/changes needed to not be so confusing, however, are still entirely valid. Sure Beae (talk) 16:00, 6 September 2026 (UTC)

The calculus of finite differences inner strip (when there are exactly 3 strips) is probably actually obtained exactly through a quadratic equation type deal: ${\frac {G\left(x\right)+F\left(x\right)-\operatorname {sign} \left(\sin \left(2\pi x\right)\right){\sqrt {\left(G\left(x\right)-F\left(x\right)\right)^{2}+4\pi ^{2}}}}{2}}$, in the general case. At least, this works for this example in desmos: https://www.desmos.com/calculator/s6nzutnaf8 Sure Beae (talk) 16:27, 6 September 2026 (UTC)

If anyone can recognize this or similar, please give me a citation so I may add it to the article. If it is original, anyone is allowed to take it without attribution (everything I say is CC0) and write about it. If you do turn it into a paper, please give me the title/doi/citation so I may add it to the article. Sure Beae (talk) 16:35, 6 September 2026 (UTC)

I couldn't think of any smart way to cheat the pole structure through calculating it (I think it's halting problem equivalent if thin strips are involved) using Abel-Plana or any sort of numerical tricks, so yes, ${\frac {G\left(x\right)+F\left(x\right)\pm {\sqrt {\left(G\left(x\right)-F\left(x\right)\right)^{2}+4\pi ^{2}}}}{2}}$ is probably the best I can provide in this case. Sure Beae (talk) 18:45, 6 September 2026 (UTC)

Which is the general way for this specific type of problem (type of $f$ in the limited, real axis only pole set with $2$ unique poles) because of the periodic nature allowing you to use trig identities. This does not work in general. You would need a priori knowledge of pole structure to create a real truly general formula/algorithm for any two unique splitting real parts, as far as I am aware. Sure Beae (talk) 01:52, 22 September 2026 (UTC)

GA review

The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

---

This review is transcluded from Talk:Indefinite sum/GA2. The edit link for this section can be used to add comments to the review.

Nominator: Sure Beae (talk · contribs) 04:12, 12 July 2026 (UTC)

Reviewer: David Eppstein (talk · contribs) 00:10, 28 July 2026 (UTC)

Quickfail WP:QF #2, #5: many unreferenced paragraphs, as noted in the previous review:

• Section "Forward and backward difference conventions", missing citation for material from "analytic continuation" to the "discrete counterpart" displayed equation (one or two paragraphs)
• Section "Fundamental theorem", missing citation for "Alternatively, using the inverse backward difference operator" (one paragraph)
• Section "Falling factorials", missing citations for most of the boxed example (multiple paragraphs)
• Section "Summation by parts", missing citations for second part of product rule, symmetrical form of summation by parts, and almost all of the example (multiple paragraphs)
• Section "Uniqueness of the principal solution", missing citations for entire first paragraph
• Section "Complex analysis (exponential type)", missing citations for end of third paragraph (Nørlund’s theory), end of fourth paragraph ("it is a matter of convention"), end of fifth paragraph ("Alternatively, authors may designate"), 2nd paragraph of proof sketch, end of 3rd paragraph of proof sketch, and several paragraphs in hidden "Example: Partitioning disjoint connected components"
• Section "Müller–Schleicher axiomatic method", missing citations on the six axioms and on several paragraphs at the end of the section

This was only roughly halfway through the article; at this point I gave up listing examples of uncited material. This was not ready for a renomination after Talk:Indefinite sum/GA1. —David Eppstein (talk) 00:10, 28 July 2026 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Multiple issues requiring revision

I think this article needs substantial review and restructuring.

• The article combines Nørlund summation, convexity methods, Abel-Plana, Euler-Maclaurin, special functions, queueing theory, compiler optimization, etc. into a unified picture of "indefinite summation". Per WP:SYNTH and WP:NOR, this overall framework should itself be supported by reliable secondary sources, rather than constructed by combining separate sources. Otherwise, the affected portions of the article should be removed or rewritten.

• Large parts read more like an extended exposition or set of notes than an encyclopedia article, with long derivations and extensive treatment of particular methods. This raises concerns under WP:NOTTEXTBOOK and WP:UNDUE, particularly regarding the prominence given to Nørlund's method.

• The article describes extending values defined on the integers to real or complex arguments as "analytic continuation". This is an interpolation problem: analytic continuation requires an analytic function already defined on an open domain. This distinction should be corrected.

• The infobox says the domain is functions on the integers while also saying solutions are unique up to an arbitrary 1-periodic function. On $\mathbb {Z}$, a 1-periodic function is simply constant.

• The notation $\Delta ^{-1}$ is not in itself problematic, but the article calls it a "linear operator that inverts" $\Delta$ while also correctly explaining that antidifferences are nonunique up to a periodic function. The article should explain how the minus one exponent is being interpreted, purely as a notation, or as a particular right inverse.

These issues also affect the lead, which currently commits strongly to one particular framework. Expert review from someone familiar with finite differences or complex analysis would be useful. Nolord (talk) 09:19, 29 September 2026 (UTC)

Thanks for the detailed review. I’m open to restructuring and help with maintenance, and I’ve already proposed a split at Talk:Finite difference#Proposal to split the article; article scope that addresses some of the scope concerns. That proposal hasn’t drawn responses yet, but it’s the direction I’d like to take. Feedback there or here is welcome. For each concern,

• Synthesis / NOR

I agree the article mixes sources into a unified framework. I wanted a theory article instead of a list, so I wrote a compendium, and it should be fixed to reflect Wikipedia better (I don't know how to do this well; the subject is complicated and difficult to communicate). The calculus of finite differences is a recognised field per Boole, Nørlund, Jordan, Milne-Thomson and it should get its own article. I’ve proposed splitting the applications (queueing, compilers, Casimir) into a separate article or trimming them. I think the core in depth coverage of the theory should stay under "indefinite sum" or be moved to "antidifference" to avoid preconceived notions of what the article "should be" that I've experienced (see Article scope on the archive 1). Help with that split would be welcome. The amount of work that needs to be done in my proposal is too much for me to complete without getting burnt out partway.

• NOTTEXTBOOK / UNDUE

I agree the article is long and could use trimming of derivations. But I don’t agree that Nørlund’s weight is the problem. Nørlund’s Vorlesungen über Differenzenrechnung is the foundational text of the field, and he is cited by essentially everyone working in it. Müller–Schleicher, Marichal–Zenaïdi, and Candelpergher are working within, or rediscovering, results that are already in Nørlund. Framing the modern papers as the novel contributions and Nørlund as one school among many would be less faithful to the literature, not more. The UNDUE concern, if any, is in the applications sections I wrote, not in the theory, when theory is the intended direction I had when I inherited the list version of the article. I’m happy to trim those and move derivations into collapse blocks, but I’ll keep Nørlund as the central source because most of the books in the Resources topic reference him or a direct derivative source that cites him.

• Analytic continuation vs. interpolation

I agree with the specific point that extending a discrete sum from the integers to real/complex arguments is interpolation, not analytic continuation. There are two distinct operations in the article:

- Nørlund’s Analytische Fortsetzung-extending the Hauptlösungen between pole free vertical strips through recurrence (as well as what Ramanujan and Candelpergher do)-is genuine analytic continuation. The functions are already analytic on an open domain, so the term is correct there. Renaming it to “analytic solution” would invent terminology and lose Nørlund's. (I was once advised to do this by email, and traces of that usage are still in the article; I or someone else should remove/clarify them.) I stand by the current translation and am opposed to changing it.

- Extending a discrete sum to non-integer arguments is interpolation. That is the operation that should be relabelled.

The lead and other prose need better clarity on this. I'm working alone, and copy editing my own prose repeatedly is laborious.

• Infobox

Done. Changed the infobox to "principal solutions of difference equations, analytically continued between maximal singularity-free vertical strips," which removes the integer domain inconsistency and uses analytic continuation in Nørlund's sense.

• $\Delta ^{-1}$ notation

I agree that it could be explained better, but it is the standard notation. The less standard notation is $\nabla ^{-1}$, which I inherited from the earlier version.

• Splits & new work

I’ve proposed a three way split at Talk:Finite difference#Proposal to split the article; article scope:

- New Calculus of finite differences for operator theory, indefinite sum/product overview (like this article's coverage of indefinite products).

- Trimmed Finite difference as a hub.

- Numerical material moved to Finite difference method.

I also noted scope overlap with Discrete calculus at Talk:Discrete calculus#Scope creep. That article mixes elementary discrete calculus and algebraic topology; the analytic operator theory belongs with the new article, leaving the topological content in Discrete calculus.

I’m somewhat burnt out, so help with the restructuring and with maintaining the article(s) would be appreciated. If anyone wants to collaborate on the split, please respond here or on the proposal talk page topic. Sure Beae (talk) 20:11, 29 September 2026 (UTC)

• Is "calculus of finite differences" actually established as a subject distinct from "finite differences"? As far as I know, the latter is simply the short form of the former.
• If no authoritative source establishes a theory connecting the various fractional/extended summation methods, Wikipedia should not create one. The article should instead list the independently sourced approaches. I think the article would also benefit from this change in clarity.
• As a side note for when the article is in better shape, two additional elements of such a list come to mind: Alabdulmohsin's "fractional finite sums" and the Borel-Laplace approach in resurgence theory (see e.g. Corollary 5.11 in Divergent Series, Summability and Resurgence I: Monodromy and Resurgence).
• The lead should mention "fractional sum" as an alternative common term used in the literature.
• I agree that the numerical-method material should be moved to the corresponding finite difference method article.

Otherwise, I leave the remaining content issues to your judgement. Nolord (talk) 01:06, 30 September 2026 (UTC)

"Is 'calculus of finite differences' actually established as a subject distinct from 'finite differences'?"

Yes, and the distinction is visible in the titles of the major treatises:

• George Boole, A Treatise on the Calculus of Finite Differences
• N. E. Nørlund, Vorlesungen über Differenzenrechnung (1924)
• Other books titled Differenzenrechnung, such as A. A. Markoff (German translation)
• L. M. Milne-Thomson, The Calculus of Finite Differences
• Charles Jordan, Calculus of Finite Differences

It's mostly in the naming of the textbooks themselves as well as their contents. Steffensen's Interpolation doesn't cover the analytic theory. It's a good source on the strict interpolation side, but nothing more.

These are the works that treat the analytic theory of the difference operator: the antidifference, Bernoulli and Euler polynomials, generating functions, factorial series, and the connection to special functions. The central object-the antidifference-is ambiguous up to an arbitrary 1-periodic function, so analysis (growth conditions, minimal exponential type, contour integrals) is required to fix a canonical solution. That is what "calculus of finite differences" names, and it is distinct from "finite differences" in the numerical methods sense (difference quotients, stencils, error orders, finite difference methods for PDEs), which has its own separate field.

There is also a parallel historical tradition. Ramanujan worked on the same underlying problem-summation of divergent series and its relation to special functions-using different, independently discovered techniques. His notebooks on the subject are extensive. Nørlund's book was published in 1924; Ramanujan died in 1920, and there was no direct interaction between the two traditions. Both treat the same subject from different angles, and the modern literature-Candelpergher's lecture notes and recent paper, for instance-is still making the connections.

So yes, the subject is real and has a long history, and it's related to resummation. It's more of a sister to Nachbin's theorem and Mittag-Leffler summation than most else. But it's more of an umbrella term that also encompasses the finite difference literature. Its main topic is on the formal series and functional side, including interpolation and analytic continuation, when referenced alone. But it has been and can be used as an umbrella term.

"If no authoritative source establishes a theory connecting the various fractional/extended summation methods, Wikipedia should not create one. The article should instead list the independently sourced approaches."

On "list the independently sourced approaches": the analytic theory of the difference operator is not a synthesis I constructed. Nørlund's Vorlesungen über Differenzenrechnung is the source that unifies the summation problem, the principal solution, the Abel–Plana formula (via him and Candelpergher), factorial series, and the applications to special functions. Müller–Schleicher (already cited in the article, and covers the exact fractional summation) and the modern papers work within or adjacent to his framework; the works often cite him. Treating his theory as one method among many would misrepresent the literature. The WP:SYNTH concern would apply only if I had built the unification myself; I did not. See also the Resources topic on the talk page for the broader source list.

The literature is not fully ignorant of Nørlund. The modern papers are reinventing it, in their own ways, using telescoping and real analysis. They often cite him. The modern papers have extended the theory in particular directions (fractional orders, real-analysis methods, resummation) but have not reproduced the full scope of Nørlund. He covers more operators than they do-a mean operator for $\nabla$, an alternating operator, continued fraction functions, and much more. This is not a synthesis I invented; it is the subject as the classical literature defines it.

I have mentioned resurgence theory on this talk page. I am interested in it, and have read some of J. Écalle's Guided tour through resurgence theory, though I cannot find the link right now. If you have sources, please add them to the Resources topic and we can discuss whether it belongs in the article or elsewhere-or not on Wikipedia at all, if the sources do not state the synthesis explicitly. So far, nothing I have seen does, and if it doesn't quite explicitly, we shouldn't add it.

"The lead should mention 'fractional sum' as an alternative common term used in the literature."

That is a related but narrower question. We can split the Müller–Schleicher material into its own article if the conflation causes too much confusion. Fractional sum methods cover a subset of what the analytic theory of the antidifference operator treats, and the modern papers on them are working within or adjacent to the classical theory rather than replacing Nørlund's full theory.

On the lead rewrite: I would like to restore the earlier version. The rewrite lists $\Sigma$ and "fractional sum" as if they were standard synonyms, and it drops the interpolation vs continuation distinction. I am glad to work on length, but not at the cost of accuracy. If you have specific objections to the earlier wording beyond length, I am glad to address them.

I like the hatnote you added to the top-"Not to be confused with summation for addition of terms between bounds"-but the lead rewrite as it stands misrepresents the literature. Sure Beae (talk) 07:58, 30 September 2026 (UTC)

I am not convinced on two points:

• The cited books establish that "calculus of finite differences" is a traditional name for the broad theory of finite differences, but not that it is a subject distinct from "finite differences". The numerical-analysis meaning you describe is more specifically the finite difference method.
• Likewise, as far as I understand, what the Müller-Schleicher method achieves is not fundamentally different from what Norlund does. Even if their construction can mathematically be subsumed under Norlund's theory, you should not treat it that way unless reliable sources explicitly establish that relationship. And of course, including that classification in the article would itself be WP:SYNTH.

Nolord (talk) 08:31, 30 September 2026 (UTC)

I rewrote the lead to discuss the more central aspects of the sum in less detail, added the mention of fractional sums, the $\Sigma$ notation, and removed the corresponding issue. I think most of the citations there could be removed, or replaced by only one (and that is maybe the case elsewhere in the article).

Couple of things that are obviously missing:

• The indefinite sum evaluated at negative integers.
• A mention of the application of the indefinite sum after the forward difference, extending the telescoping sum, which is the true analogue of the fundamental theorem of calculus.

A history section would also be interesting, the first use of $\Sigma$ traces back to Euler who used it specifically as an indefinite sum operator in 1755, which is the ancestor of the bigger $\sum$ now used for the distinct purpose of summation between bounds (I am adding a disambiguation for that too). The latter apparently traces back to Fourier in 1820. See the Summation page, which could also use an edit and link here.

Also please consider which of the denominations (indefinite sum vs antidifference) and symbols are most common in the literature and consider moving (renaming) the article to antidifference (as was discussed here) and adopting a different notation (or not). Nolord (talk) 05:25, 30 September 2026 (UTC)

I disagree with the rewrite because the subject is not equal to fractional summation. Please read the above reply.

It is solving difference equations with minimal exponential type, dependent on the native pole free maximal vertical strip of the complex plane. An inner solution does not equal a telescoping from either side.

As for renaming, I'm all for renaming it to Antidifference or Indefinite sum (analytic continuation) to reduce confusion. It is, however, more common in the literature to use "indefinite sum". I've always been undecided on this. Sure Beae (talk) 08:05, 30 September 2026 (UTC)

I think there is a distinction being blurred here between an indefinite sum itself and Norlund's principal-solution theory. An indefinite sum is, at the basic level, simply a solution of the difference equation, while Norlund's theory imposes additional analytic or growth conditions to select a distinguished solution. So when you describe the subject as solving difference equations with minimal exponential type on a maximal pole-free strip, that sounds to me like a description of Norlund's principal-solution theory rather than of the general concept of an indefinite sum.

Could you also explain what distinction you are making between an indefinite sum and a fractional sum? I do not currently see where one ends and the other begins. Nolord (talk) 08:32, 30 September 2026 (UTC)

https://www.desmos.com/calculator/rcqydruzkf

The inner solution (which exists per Nørlund) cannot be reached by the telescoping described by Müller–Schleicher. Müller–Schleicher and fractional summation in general is also limited in span/step size $=1$ as far as I recall. Sure Beae (talk) 08:39, 30 September 2026 (UTC)

So you are saying that Norlund's work is far more general, but not that the underlying concept is fundamentally different. "indefinite sum" "antidifference operator" and "fractional sum" designate a concept not a specific theory. And to me, everywhere they are used, they designate the same concept. Nolord (talk) 08:51, 30 September 2026 (UTC)

Nørlund covers all span sizes and solutions which the real analysis and Müller–Schleicher methods cannot. The Müller–Schleicher method is limited in its possible to reach exponential type and unrelated in general to Abel-Plana. They use polynomials/monomials to construct their argument. They are all valid theories, and they describe the same fundamental objects of a solution to forward or backward difference equations. However, each method is different and describes different but related things, handling the $C(x)$ term differently. I believe calling things other than what Nørlund, Milne, and Jordan present "indefinite sum"s is inaccurate.

I am on chimera linux. There is no https://github.com/phiresky/ripgrep-all package. Are you on an operating system where it is packaged/binaries are available? Most of these books are in the public domain or the papers are free and can be downloaded. We can easily answer the usage by grepping for it. Sure Beae (talk) 09:04, 30 September 2026 (UTC)

Is that Linux question a mis-paste? if so please remove it as well as this comment, because I don't see the relevance. Nolord (talk) 09:45, 30 September 2026 (UTC)

No, it was an invitation to you to find sources? Sure Beae (talk) 09:54, 30 September 2026 (UTC)

Okay, I get it now. You can use Google Scholar for that purpose. Nolord (talk) 10:03, 30 September 2026 (UTC)

It can search in the texts themselves? Does it catalogue our older Resources topic entries? Sure Beae (talk) 10:09, 30 September 2026 (UTC)

It can search in the text of books and papers that are available online, so generally the more recent ones. Google Books is more suited for book searches. Nolord (talk) 01:34, 2 October 2026 (UTC)

We already have basically everything needed to cite all the content in the article under the Resources topic. Your issue was mainly about splitting the different things out, due to length and synthesis concerns, no?

I don't think the issue is finding more citations, but using what is already present more often in the text, and splitting out articles to things like

Indefinite sum (real analysis)

Indefinite sum (complex analysis)

Fractional sum

Indefinite sum (applications)

etc., probably with better names. Sure Beae (talk) 10:48, 2 October 2026 (UTC)

What is your exact point, here? If you are arguing for an integer case, it's wrong. None of the papers or books support that. Please see the article's cited sources. Each paper/book either uses complex numbers/complex analysis (Müller, Nørlund, Milne, Candelpergher, ...) or real analysis (Marichal and Zenaïdi). If you have a point, please back it up with sources that counter what is already present in the article, so we may cite them.

The ambiguity of $C(x)$ is what all of the theories are about. There is no getting around it.

Can you state in math notation or writing exactly what you mean/want? Sure Beae (talk) 09:10, 30 September 2026 (UTC)

The burden of proof that fractional summation and indefinite summation (in the sense it was used in the literature) are the same needs sources. I have not seen something claiming fractional summation is as general or equal to indefinite summation, which is any solution to a difference equation. I have only really seen telescoping and related, there.

The burden of proof doesn't lie with me on this. I already had citations. Sure Beae (talk) 09:21, 30 September 2026 (UTC)

My point is about the lead, and more generally what the article should look like.

Nothing you have said shows that the current lead is inacurrate. Mentioning Norlund's theory in the lead would be like mentioning the Riemann–Liouville fractional derivative in the Fractional calculus lead. I hope that analogy is clear.

Fractional sum and indefinite sum describe the same underlying concept, approached from different motivations. That distinction should be explained in the body, but "fractional sum" should remain in the lead; otherwise the lead fails to tell readers what the article actually covers.

As for the burden of proof, I don't think a separate source is needed merely to observe that two definitions describe the same mathematical concept. Otherwise we would also need a source explicitly saying that "antidifference" and "indefinite sum" are the same whenever the sources define both by the same difference equation, and likewise for the notations.

The distinction between the motivations and the additional conditions imposed by the different theories can be discussed in the body. Nolord (talk) 09:44, 30 September 2026 (UTC)

"Fractional sum" is not an equal term to "indefinite sum" or "antidifference". You need to source that. Sure Beae (talk) 09:51, 30 September 2026 (UTC)

Please do not dismiss my arguments if you want to have a constructive discussion. It already addresses this point. Nolord (talk) 09:57, 30 September 2026 (UTC)

We can say the term is related, and not cite it. That is standard usage, that is correct. We cannot say that it's an alternative term for the exact same thing, as the literature doesn't say that. It's inaccurate.

"Indefinite sum" and "antidifference" are perfectly interchangeable even in the literature. I'm not dismissing it, I'm trying to ask you to correct it... Sure Beae (talk) 09:59, 30 September 2026 (UTC)

I agree that I cannot currently establish that "fractional sum" and "indefinite sum" are interchangeable terms in the literature. My point is not terminological. They are two literatures dealing with the same underlying problem, and the article already treats Müller-Schleicher's theory as part of this subject.

If an explicit source connecting the two lines of research is required before "fractional sum" can even be mentioned in the lead, then by the same standard I do not see what justifies the Müller-Schleicher section being in this article at all. I don't think removing it is reasonable. The sensible solution is to mention the connection in the lead without claiming synonymy, and explain the different motivations and theories in the body.

I will do the relevant edit.

The notation $\Sigma$ is perfectly standard however. Nolord (talk) 10:35, 30 September 2026 (UTC)

My issue is mostly semantic meaning of terms/usage in the literature. I want to reflect the literature's usage accurately.

My proposal on the Finite difference talk page was about addressing overlap and lack of accuracy. That article also supposedly "invents" the Finite_difference#Calculus_of_finite_differences as something separate. If you take issue with the "calculus of finite differences" meaning something different, please fix it/mention it on that article, too. We use similar meanings for that term.

My proposal wasn't strictly about this article. It was about all of the finite difference adjacent articles on Wikipedia. Sure Beae (talk) 10:56, 30 September 2026 (UTC)

"The notation Σ is perfectly standard however" can you please add the inline citation along with your notation addition?

To whatever sources used at

https://mathshistory.st-andrews.ac.uk/Miller/mathsym/operation/

Are applicable.

There is still no inline citation for it. I'm not arguing about if it's correct or not, I'm asking you to cite your additions inline rather than making me do more work. I'm not going to cite other's additions for some time.

List of indefinite sums was hard enough to find sources for, as most of it was content others had added, so I had to go looking for ways to cite it, and it's still not ideal at all.

I do not feel like chasing other people's claims for sources for the time being. Sure Beae (talk) 14:51, 1 October 2026 (UTC)

Fixed it using https://books.google.com/books?id=D7cF_pn7NhoC Sure Beae (talk) 22:02, 1 October 2026 (UTC)

You also used one of the citations I had reused from the article for the $\Sigma$ notation. On the other hand, I have not seen a secondary source that uses the $\sum _{x}$ notation. That isolated paper is not sufficient as a citation. Nolord (talk) 23:49, 1 October 2026 (UTC)

"That isolated paper" refers to what? The book link I gave is Kelley.

Jordan, L. M. Milne-Thomson, Jagerman are likely to be secondary to Nørlund and likely contain the $\sum _{x}$ notation. Sure Beae (talk) 11:06, 2 October 2026 (UTC)

It refers to your reference Man, Yiu-Kwong (1993), "On computing closed forms for indefinite summations". You can see it in the old version of the article.

No, they do not. All three use a big "S". Nolord (talk) 11:40, 2 October 2026 (UTC)

The big curly S on page 201 of L. M. Milne and in the other books is not recognizable to me as a character. I don't know how to type it in math blocks. Jagerman and the paper Sukruoglu & Simsek use small S if I recall? I tried checking the list on 107 Jordan for $\sum _{x}$ and you're right, it's not there either. He uses Delta.

What do you suppose we use? The plain letter S? Sure Beae (talk) 11:58, 2 October 2026 (UTC)

I think one of our older books definitely has the $\sum _{x}$ notation present in a list. Sure Beae (talk) 11:59, 2 October 2026 (UTC)

I don't believe I was the one to add On computing closed forms for indefinite summations. Sure Beae (talk) 12:01, 2 October 2026 (UTC)

My bad, that may not have been yours.

I am not sure we should put the "S" in the article, it doesn't look as common in journal references. And it is more complicated, because of the Delta after that mimics the differential in integrals.

But adding it is totally justifiable too. Nolord (talk) 12:14, 2 October 2026 (UTC)

"And it is more complicated, because of the Delta after that mimics the differential in integrals." I think that's actually better, for coverage of the analytic side. Maybe the telescoping side could leave it out. It closer parallels Differintegrals, and conveys that it acts upon functions/strips, not integer arguments.

We should keep the notation in each section accurate to the respective literature. I mostly inherited the $\sum _{x}$ notation, though it looks familiar to me even outside of the context of the article. I think it is possible to cite to a secondary source. Sure Beae (talk) 12:27, 2 October 2026 (UTC)

Please see the new citation neededs added and fix them. Sure Beae (talk) 22:30, 1 October 2026 (UTC)

As for restructuring the article itself in full/splitting content out, I am fine with you making changes and welcome it per the proposal. I am asking you to review the sources to not include inaccuracies in the simplified version. Sure Beae (talk) 10:01, 30 September 2026 (UTC)

I want the lead and full article to be concise and understandable, but not at the cost of being incorrect. I want to leave the hatnote, it's a good addition. I want to revert the lead rewrite. It isn't accurate. May I? Sure Beae (talk) 08:07, 30 September 2026 (UTC)

Let's clarify things first Nolord (talk) 08:36, 30 September 2026 (UTC)

I suggest removing the videos and the portrait of Norlund. The animations seems redundant with the adjacent one-line explanation per Wikipedia:Manual of Style/Images#Video content. The Norlund portrait also seems mainly decorative, see MOS:IMAGEREL and WP:IMGCONTENT. Nolord (talk) 01:29, 2 October 2026 (UTC)

I propose to reorganize the article roughly as such:

• History
• Basic example
• Choosing an antidifference operator
  • The Muller-Schleicher axiomatic method (as a conceptual introduction of the issues)
  • Norlund's principal solution (for the more complete theory, but with selected pieces of information)
  • Maybe Abel-Plana, Euler-Maclaurin, etc...
• Properties
• Indefinite product
• Examples
• Applications

Nolord (talk) 03:09, 2 October 2026 (UTC)

I don't think removal of video content would be very helpful to the reader, I am against it. I think moving videos to their respective articles is appropriate.

"The Muller-Schleicher axiomatic method (as a conceptual introduction of the issues)

Norlund's principal solution (for the more complete theory, but with selected pieces of information)"

The first is fractional summation (intent: find a continuation of summation. Result: finds a left and right summation through telescoping, which is what Muller-Schleicher (complex) and Marichal-Zenaïdi (real) did in practice), the second is analytic continuation of a finite difference equation (intent: find a distinguished solution to first order linear difference equations. Result: finds every strip, not just a left and right summation version).

Can you be more clear on what you think is synthesis? "Indefinite sum" and "fractional sum" are not interchangable, the Muller-Schleicher paper doesn't use "antidifference" or "indefinite sum" and I have not seen literature saying they are strictly interchangeable.

If you are intending to reduce synthesis, I think the correct way is to leave this article only with what comes from Nørlund, Jordan, L. M. Milne-Thomson, Jagerman, Candelpergher, Sukruoglu & Simsek, J. D. Gray, with a title like "Nørlund principal solution" or "Indefinite sum (complex analysis)". And to split the other ones into their own articles as I suggested above. Sure Beae (talk) 11:02, 2 October 2026 (UTC)

It is not about what you think is clear, it is about the guidelines MOS:IMG : Videos should be used as a supplement to article material, to concisely illustrate the subject in a way that a still image or text cannot do; they do not specifically illustrate the subject.

"Indefinite sum" and "fractional sum" are not interchangeable terms, but that does not make them unrelated: they study the same underlying concept. Creating a separate article for fractional summation would therefore be unreasonable. The difference in both motivation and method between Muller-Schleicher and Norlund is what's interesting. There is no SYNTH problem in placing them in the same section and explaining how their motivations and methods differ. Nolord (talk) 12:08, 2 October 2026 (UTC)

The video for the complex plane partitioning I believe is hard to convey without the video. The others I could see as understandable without per that rule. Sure Beae (talk) 12:28, 2 October 2026 (UTC)

"The others" refers to the videos specifically, not the images. What you have quoted represents the policy on videos. Sure Beae (talk) 20:39, 2 October 2026 (UTC)

'"Videos should be used as a supplement to article material, to concisely illustrate the subject in a way that a still image or text cannot do"; they do not specifically illustrate the subject.'

I read that as contradictory. The MOS:IMG you cited says that videos should be used when language doesn't suffice, then you continue with 'not specifically illustrate the subject'. Which is it? Or can you explain it in a way that is easier to understand? Sure Beae (talk) 12:43, 2 October 2026 (UTC)

Finite Difference Visualisation One Periodic Vanishing.webm illustrates the 1-periodic $C(x)$ vanishing under a shift by 1.

Inverse Finite Difference Partitioning.webm illustrates the complex plane with poles blocking strips/the construction of half planes for which there are unique solutions on.

Müller Digamma.webm shows more terms of the telescoping being added, which constructs digamma. Perhaps the weakest in terms of needing this level of visualization. Could be an image reasonably, per the rule.

Odd Symmetry Nörlund Principal Solution.png shows Ergänzungssatz

Analytic Continuation Harmonic.png shows the shifted digamma function, which is the indefinite sum of $1/x$.

I would like to hear the specific reason why for each of the removals. Is it the caption? The content?

"Removing videos and images not illustrating indefinite sums specifically per MOS:IMG, except one image for illustrating the introductory example. Consider ammending the image of the infobox with other interesting examples."

Is not specific enough for me to act upon it. And, well, wrong as stated because digamma and loggamma are objectively indefinite sums of $1/z$ and $\ln \left(z\right)$ respectively.

Your complete nuking of every single image/video like this seems odd. This rule doesn't justify that to my reading. And I need you to elaborate on "they do not specifically illustrate the subject". Sure Beae (talk) 20:27, 2 October 2026 (UTC)

Additionally, the removal of the Gamma and loggamma images seems odd. Those are trivial indefinite product and indefinite sum respectively. Which is directly explained in the article with inline citations. Sure Beae (talk) 20:30, 2 October 2026 (UTC)

• Finite Difference Visualisation One Periodic Vanishing.webm: MOS: video content says videos should concisely illustrate the subject in a way that a still image or text cannot do. Here $C(x+1)=C(x)$ and $\Delta C=0$ are already stated directly. Its second use also raises the MOS:PERTINENCE requirement that each image have a clear and unique illustrative purpose.
• Inverse Finite Difference Partitioning.webm: WP:IMGCONTENT says images should increase readers' understanding of the article's subject matter, and MOS:PERTINENCE requires them to be significant and relevant in the topic's context. This illustrates an auxiliary partition of the complex plane, rather than indefinite summation itself. Moreover, MOS: video content says video should illustrate the subject in a way that a still image or text cannot do. It is unclear what motion contributes to illustrating this static partition.
• Odd Symmetry Nörlund Principal Solution.png: the symmetry is already stated exactly as $F(z)=F(-1-z)$. WP:IMGCONTENT says The relevant aspect of the image should be clear and central. This adds little beyond illustrating a subsidiary property.
• Analytic Continuation Harmonic.png: MOS:PERTINENCE requires a clear and unique illustrative purpose and an important illustrative aid to understanding the subject. It is unclear how a generic plot of a shifted digamma function illustrates the normalization condition being discussed.
• LogGamma Analytic Function.png and the 3D Gamma plot: these illustrate $\log \Gamma$ and $\Gamma$ themselves, rather than the sum–product relation being discussed. WP:IMGCONTENT states that The purpose of an image is to increase readers' understanding of the article's subject matter; it is unclear how these generic function plots aid understanding of indefinite summation specifically.

The issue is not whether the media are related somehow, but whether they materially aid understanding of indefinite summation. MOS:PERTINENCE also notes that usually, less is more.

If this is reverted again, please give a substantive, policy-based justification addressing these points. I don't think removal of video content would be very helpful to the reader, I am against it does not do so. WP:CONSENSUS explicitly states that The arguments "I just don't like it" and "I just like it" usually carry no weight whatsoever. If a particular image or video should remain, please explain what clear illustrative purpose it serves under MOS:PERTINENCE and, for video, what it communicates that text or a still image cannot under MOS: video content. Nolord (talk) 23:44, 2 October 2026 (UTC)

"When possible, find better images and improve captions rather than simply removing poor or inappropriate ones, especially on pages with few visuals." Sure Beae (talk) 23:51, 2 October 2026 (UTC)

Under Pertinence and encyclopedic nature in MOS:IMG. Sure Beae (talk) 23:53, 2 October 2026 (UTC)

I agree that replacing an unsuitable image with a better one is preferable when possible. If you have more pertinent images or better captions in mind, please propose or add them. I do not have suitable replacements, and relevant images can of course be added later. Nolord (talk) 23:57, 2 October 2026 (UTC)

Most of the images and all of the videos on this page were made by me. I could replace them myself if you were cooperative and not constantly jumping the gun. This has become stressful for me, and I am burnt out, so I am stepping away. Goodbye for a few months. Sure Beae (talk) 00:03, 3 October 2026 (UTC)

Understood. I did not intend to make this stressful for you. I appreciate that you made much of the media and were willing to replace or improve it.

I won't seek a third opinion while you are unavailable. I do intend to continue improving the article in the meantime, including making the media changes. Relevant replacement media can of course be added later.

When you return, you are still welcome to discuss or challenge any of the changes I have made in the meantime. Nolord (talk) 00:30, 3 October 2026 (UTC)

If you continue to try to smuggle even more statements to pages that do not contain them, or use notation for which a source does not contain in that exact way, I will end up reverting them/flagging with citation neededs/having to rewrite the notation and find a proper source after I come back. Do not make considerable work for me, or I will never return. I do this for fun, and find cleaning up after others very not so. Sure Beae (talk) 02:54, 3 October 2026 (UTC)

Receipt: https://en.wikipedia.org/w/index.php?title=Indefinite_sum&oldid=1378129770 Sure Beae (talk) 02:55, 3 October 2026 (UTC)

I will do my best while following the manual of style. For the time being, I'll be the one cleaning up. Nolord (talk) 03:43, 3 October 2026 (UTC)

"per WP:NOR Rewriting source material in one's own words while retaining the substance is not considered original research. and Source material should be carefully summarized or rephrased without changing its meaning or implication"

It is at the least synthesis of fractional summation and indefinite summation, conflating the two. It is not accurate or even relevant to the contents of that page, it states the end of an example, the right inverse, the \sum notation (it is not a lowercase sigma [edit: small capital sigma, it is late and I am tired] and combining sources to new notation in this way is synthesis), then provides an integral.

"For positive integer arguments, the indefinite sum reduces to ordinary summation." Is not supported by it at all. You are synthesizing. Seek third opinion, RfC, whatever you want. I'm done. You are filibustering. Sure Beae (talk) 03:43, 3 October 2026 (UTC)

And edit warring. Sure Beae (talk) 03:44, 3 October 2026 (UTC)

The lead is precisely not conflating fractional and indefinite summation, changing notation is not synthesis and see Theorem 2.7 for integer arguments. Nolord (talk) 03:47, 3 October 2026 (UTC)

That doesn't fix my concern. Theorem 2.7 is on page 27. Not page 20. The inline citation is not accurate. I am not going to clean up after you. Sure Beae (talk) 03:50, 3 October 2026 (UTC)

Yes it's fixed now. Nolord (talk) 03:53, 3 October 2026 (UTC)

Thank you. I really will not come back if it is in an even worse state than I left it at citation wise. It brings me too much stress and anxiety. Sure Beae (talk) 03:56, 3 October 2026 (UTC)

'Inverse Finite Difference Partitioning.webm: WP:IMGCONTENT says images should "increase readers' understanding of the article's subject matter", and MOS:PERTINENCE requires them to be "significant and relevant in the topic's context". This illustrates an auxiliary partition of the complex plane, rather than indefinite summation itself. Moreover, MOS: video content says video should illustrate the subject "in a way that a still image or text cannot do". It is unclear what motion contributes to illustrating this static partition.'

That would be valid under my reading if you also modified the text to be more clear on the partition for those with a screen reader. These rules seem to be written with accessibility in mind. Sure Beae (talk) 23:58, 2 October 2026 (UTC)

Analytic Continuation Harmonic.png and LogGamma Analytic Function.png are simple examples of their respective sections. What about the captioning or surrounding text makes that unclear such that they need removed? Sure Beae (talk) 00:01, 3 October 2026 (UTC)

I don't think further point-by-point back-and-forth between the two of us is likely to resolve this. I'll seek a third opinion on the remaining disputed media. Nolord (talk) 00:08, 3 October 2026 (UTC)

I have removed the Nørlund portrait and Müller video per MOS:IMG. Please explain the rest and come to consensus before making a similar bold edit. Sure Beae (talk) 20:47, 2 October 2026 (UTC)