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)

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)

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)

Further lead improvements for accessibility

See:

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

The earlier discussion about article scope highlighted that the lead could be made more accessible to readers who expect a strictly discrete topic. The article now makes clear that the indefinite sum solves a first-order linear difference equation and is a continuous operator. That won't (or rather, can't) change (Wikipedia verifiability policy, no original research policy), because it's what the sources say.

I'm interested in suggestions that could make the lead softer to newcomers, or help connect to content that is widely known (in a way that's allowed/aligned with our sources). The Forward and backward difference conventions section and Examples section are designed to connect this, but the lead and those sections can be restructured and rewritten to improve pedagogical clarity for new readers. Sure Beae (talk) 18:37, 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)

Real analysis section needs visualizations

The Real analysis (higher-order convexity and concavity) section is closely related to the Müller–Schleicher method (essentially a telescoping sum as defined in the Müller–Schleicher paper with a Newton series remainder). Michael DiFranco (Lines That Connect) derived the same formula independently in a video here (stated for imaginary inputs $x+iy$ and then for real inputs $x$):

$\lim _{x\to \infty }\Delta ^{m}f(x)=0,$

$S(x)=\lim _{n\to \infty }\left(\sum _{k=1}^{n-1}(f(k)-f(x+k))+\sum _{k=1}^{m}{\binom {x}{k}}\Delta ^{k-1}f(n)\right),$

and Markus Müller acknowledges the connection on his website. DiFranco’s Manim community code for that video is at GitHub for reference. This is essentially identical to Marichal and Zenaïdi's results in their paper and book. In the video for Müller's method I already show the independent components contributing for the inverse backward difference of $1/x.$

Right now the section has no visual aids, and higher-order convexity/concavity is difficult to come up with unique visuals aids for. I’d appreciate to hear ideas for graphical illustrations that don't mimic the current images/videos. Any suggestions are welcome/would be appreciated. Sure Beae (talk) 18:28, 8 July 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/GA1. The edit link for this section can be used to add comments to the review.

Nominator: Sure Beae (talk · contribs) 05:32, 11 July 2026 (UTC)

Reviewer: Phlsph7 (talk · contribs) 08:28, 11 July 2026 (UTC)

Hello Sure Beae and thanks for all your improvements to this article. However, despite the improvements, the article fails criterion 2b since there are too many unreferenced paragraphs. Examples are the paragraphs starting with "Falling factorials provide the discrete analog of the power", "Indefinite summation by parts is the discrete analog of integration by parts", "The usual formulation assumes that the summand", and "To match this convention, the constant is fixed by requiring that the". According to criterion 2b, these and similar passages require inline citations "no later than the end of the paragraph". I suggest that you add all the relevant references before a renomination.

A few other observations

• WP:EARWIG detects no copyvios
• functions by imposing restriction on their growth replace "restriction" with "restrictions"
• by the step length every neighbouring component I think some kind of preposition is required before "every"
• singularities (e.g. logarithms, positive fractional powers), indefinite sum may merge should there be an article (an/the) before "indefinite"?
• the crucial axiom which allows one to replace "which" with "that"
• is a entire function replace "a" with "an"
• used in context of Bernoulli add "the" before "context"
• generalized harmonic number function, should this be "the generalized harmonic number function" or "generalized harmonic number functions"?
• under condition that add "the" before "condition"
• possible to antidifference numerically should this be "to antidifferentiate"?
• it admits to the same expansion remove "to"
• Generally speaking, the article is probably too technical: it may be acceptable for an encyclopedia of mathematics, but Wikipedia is a general encyclopedia.

Phlsph7 (talk) 08:28, 11 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.

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)

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.

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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.