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• CommentAuthorAnixx
• CommentTimeMar 6th 2012

I wonder should I explain in any question or answer what is polygamma function, what is discrete integral and other basic things?

It seems many people expressed disappointment that I did not explain them. For example, many people did not understand this answer

http://mathoverflow.net/questions/41011/what-is-the-indefinite-sum-of-tanx/42903#42903

because it included digamma function (wikipedia aticle about it being linked in the anser though). Many even downvoted my answer. It only became clear to some after another user explained my answer in easier language including the definition and basic properties of digamma so that his answer became accepted.

This situation is counter-intuitive for me given this is mathematical site. I also wonder whether I should demand explanation of definitions from any other question and answer on this site and downvote otherwise.
1.

Dear Anixx,

Mathematics is a very broad subject with many subfields.

The area of classical analysis that you work in is only a very small niche occupied by a very small fraction of mathematicians. In fact, given contemporary taste, it is quite far off the beaten track, so the fraction of mathematicians that know it is even smaller than for most subfields.

I would estimate less than 1% of mathematicians know what a polygamma function is. I haven't asked, but I would expect no mathematicians in my department of roughly 15 know what it is.

You also have to consider the population of MathOverflow. Algebraic geometry and logic are quite highly represented, whereas there are much fewer people from other areas. This means many readers of MathOverflow understand what a Deligne-Mumford stack is, considerably more than will understand what a polygamma function is.

• CommentAuthormarkvs
• CommentTimeMar 6th 2012

@Alexander Woo: Every mathematician should know what polygamma function is. In fact somebody who does not know what it is, and is not professional enough to at least look up the definition in google, cannot be called a mathematician. This has nothing to do with areas of math, etc. It is basic math which everybody in the profession must know. I am not defending the question or the OP, by the way. I did not even see the question.
2.
I don't think every question needs to define or link to basic terms. However if you want to get upvoted or avoid being closed by 3K users it is useful to have a feeling for who tends to read this site and what they might know. For example if I ask whether J=D in all regular semigroups that do not contain the bicyclic semigroup, then this question can be understood by all semigroup theorists (and answered). But to most people on the site, it will appear gibberish and will likely be closed as too localized or off topic. If I provided definitions and background the question might survive.
3.

@markvs:

The sum total of my analysis background is one semester of undergraduate real analysis, one semester of undergraduate complex analysis, and one semester of point set topology. I have never actually learned the definition of the gamma function (though I know it is some holomorphic function which specializes to factorial at integers and satisfies some simple functional equation that I have forgotten).

I never studied multivariable calculus; everything I know about the subject I learned while learning about differentiable manifolds. I never studied differential equations; everything I know about the subject comes from the half dozen or so times I have taught the introductory undergraduate course on the subject at various places.

I am not proud of my lack of background, but I am not particularly embarrassed by it. My lack of background is uncommon, but it is shared by a nontrivial number of US trained algebraists and combinatorialists. (Surely it says something that my then department chair let me teach differential equations the first time!)

In the other direction as it were, a university near mine with a mathematics PhD program offers only one graduate course in algebra. It has no algebraists on its faculty. I know this because they send their students here if they need or want to study more algebra. Students there regularly get PhDs without knowing what a module is. I believe some of these students go on to become mathematicians of good repute in their subfields without every learning what a module is.

Keep in mind that less than a third of my undergraduate education was on mathematics. In fact, at my previous job, students were forbidden from devoting more than 35% of their undergraduate studies to any single subject. (I think this is a good policy at least a certain types of schools.)

• CommentAuthormarkvs
• CommentTimeMar 7th 2012

"I am not proud of my lack of background, but I am not particularly embarrassed by it. "

That is the problem (lack of the sense of shame). A plumber who does not know basics of his profession would not announce it in front of 10000 people.
4.

Well, I don't think Alexander, or indeed anyone, should be embarrassed by what they don't know. None of us knows more than a tiny portion of all the mathematics there is.

Being too embarrassed to admit one's ignorance is a major problem in mathematical culture. I would prefer to be supportive of people who say openly that they don't know stuff, rather than try to shame them. It takes bravery to be publicly honest about one's limitations.

5.

Yes. Please define (or provide reference) for such things as $\sum_x f(x)$ when you use them in your question.

6.

@Mark: Cut us a little slack.. At least we're not this guy: He-who-I-will-not-name

7.
I think the answer here is a pragmatic one, based on the community which is present at MO. You should explain your answers and questions more fully, partly by giving more definitions, but also but paying more attention to issues of convergence and precise statements.

You seem to have learned a great deal of classical analysis, particularly related to extending concepts defined at integers to noninteger values. My vague impression of this sort of thing is that there are a lot of beautiful formulas which were worked out in the 19th century, but also a lot of imprecision and that I haven't seen a presentation in the modern language of analysis. I think that, fair or not, a lot of mathematicians have this belief. In particular, there seems to be no one on MO who is familiar enough with the concepts you are using, and comfortable with the way you are discussing them, to usefully engage with you. I was most involved in the discussion at http://mathoverflow.net/questions/17614/solving-ffxgx (meta thread http://tea.mathoverflow.net/discussion/742/why-my-answer-to-this-question-was-deleted/ ). That answer was frustrating not because of any vocabulary issue but because of the statement that "It is possible to find a half-iterate of any function following this formula ... This does not always converge, but sometimes it does." That's the sort of statement which will set off the hackles of any modernly trained mathematician, who is used to asking where, when and how something converges. Ramanujan or Euler could write this sort of thing, but mathematics has benefited tremendously by insisting that such statements, wherever possible, be made precise. I have only looked quickly at your most recent answers, but the difficulties seem to arise because you are evaluating a digamma function at |q|=1, which is normally outside the radius of convergence. I am willing to believe that your expression has a meaning, but it is not obvious to me what it is.

I want to emphasize that all of this is somewhat a practical issue based on the background of your audience. In many fields of math, the ability to have a profitable discussion among experts relies on glossing over points that all experts understand. For example, I have the vague impression that symplectic geometers have not actually found a rigorous definition of the Fukaya category, and certainly whenever I try to talk to them they quickly start throwing in a lot of caveats about how nothing they are saying is exactly right. I could be obnoxious and go into the threads about the Fukaya category and start pointing out ways in which statements that are being made there don't agree with my limited understanding of the Fukaya category, but I don't, because it is obvious that there is a community there who have enough common vocabulary to talk to each other, even if not everything they say is perfectly precise.

You have not found an analogous community here. But I think you could if you made more of an effort to explain the concepts you are using. I would focus less on defining nouns and more on defining verbs : When you say that a sum converges to something, in what sense is this convergence? If you are evaluating a function at some nonstandard argument, what does it mean to evaluate it there?

I will point out that Noam Elkies, Henry Cohn and Will Jagy, at a minimum, seem to know a lot about special functions and classical analysis, and I've seen some of the ideas you are interested in. So I think MO could be a good place to discuss these concepts, and I would like it to be. (Apologies to whatever obvious expert I didn't think of when I thought of those names.)
8.
@markvs:

The intent of this comment is just to point out a cultural difference. I don't want to argue whether you or I are wrong; I just want to point out that there is a different way of looking at this.

Frankly, I am much more embarrassed that I know basically no art history(*1) than that I know relatively little analysis. I probably find my relative lack of knowledge of analysis - at least I know something - less embarrassing than my never having read a word of Aristotle(*2).

I am a professional academic specializing in mathematics (you can say that I really specialize not in mathematics but in algebra/combinatorics if you want), and I think that professional academics in a Western society should know at least a little about all fields of Western culture, and knowing a little about art history and Aristotle is part of the basics of my profession.

I do think that it's a big problem that so many academics feel no shame about not knowing the basics of their profession (including quite many people in other fields who unfortunately know nothing about mathematics).

(*1) Art history is not arbitrary here - it is actually the field I am probably the most ignorant about - and I am not completely ignorant since I do know quite a bit of music history and religious history as well as the basic timelines of European history - of course I know nowhere near as much as even an advanced undergraduate in those areas.

(*2) I have read at least one book by each of Plato(*3), Hume, Kant, Nietzsche, and Wittgenstein, as well as books by some other less significant philosophers.

(*3) I am aware of the historical problems of talking about books(*4) by Plato.

(*4) This is just to be silly to add footnotes to footnotes to footnotes, but, in case you don't know, the book format was an invention of the early Middle Ages; the ancient Greeks and Romans wrote on scrolls.
9.

@David Speyer: Great comment. Thanks for taking the time to write it.

• CommentAuthorAnixx
• CommentTimeMar 7th 2012 edited

@David Speyer: about which answer do you speak here? It seems you are completely wrong and spreading FUD.

> I have only looked quickly at your most recent answers, but the difficulties seem to arise because you are evaluating a digamma function at |q|=1, which is normally outside the radius of convergence. I am willing to believe that your expression has a meaning, but it is not obvious to me what it is.

What is "q" here? Digamma is a holomorphic function defined at the whole complex plane, and not converging only at negative integer argument and zero.
10.
I as wrong to use the word recent, I apologize for getting the timeline wrong. I was looking at http://mathoverflow.net/questions/41011/what-is-the-indefinite-sum-of-tanx/42903#42903 and http://mathoverflow.net/questions/71643/are-these-two-functions-equal , following links from the "spam" thread, and I missed that these were not recent at all. I also should have said "$q$-digamma" and not "digamma", and should have been more careful about blending questions from one thread into the other.

In http://mathoverflow.net/questions/71643/are-these-two-functions-equal , you refer to \psi_{e^{2i}}. The subscript on \psi is, I believe, the parameter called q. My understanding (and, from the comments Gerald Edgar's understanding as well) is that this doesn't make sense when q is on the unit circle, as e^{2i} is.

The other question http://mathoverflow.net/questions/41011/what-is-the-indefinite-sum-of-tanx/42903#42903 also concerned the digamma function, but did not have q's in it. Here you gave an answer which, in the end, contained a lot of merit. But it was very hard to dig through both because, at first, there was no definition of the terms involved and because it was not clear that the formula you wrote down actually expressed a meaningful function.

Hope that clarifies things.
• CommentAuthorAnixx
• CommentTimeMar 7th 2012 edited

The "lack of the definition of the terms involved" is exactly what I am discussing in this thread. The only term introduced in that answer was the digamma function and I really wonder whether should I explain such "terms" on a mathematical forum. The wikipedia article was linked. I wonder whether I also should give definition of Gamma function it I used it? Maybe also definition of sine?
11.

Gosh, I should turn in my analyst's badge. I never heard of the polygamma function before looking at this thread (during an afternoon off at an analysis conference in Banff) despite my 44 years in the profession. Now that I have learned what it is, I will no doubt forget the definition within an hour or so but hope that it resurfaces in my memory in case I should ever have need of it.

12.

+! Bill Johnson. (link for you, Anixx)

Anixx, you should at least define symbols. That the polygamma function is denoted by a psi and not a gamma (or Gamma) would never have occurred to me without following a non-self-explanatory link and reading a paper/page (I didn't know you were linking to a wikipedia page, I thought it was an article). The FAQ asks that questions are pretty much self-contained. Merely saying at the beginning of the post "Let psi^(a)(x) be the polygamma function, where x takes values in (blah) and a takes values in (blah), please see this paper (link) for background." would have saved a lot of trouble. I hope that this helps you in further questions. I think it is unfortunate that people seemed to have jumped on some sort of down-voting bandwagon for the question that started this.

13.

Bill: lol! I wonder if there are Fields Medalists out there who should be handing back their medals.

14.
Thanks to Bill Johnson for making me feel less ashamed. I'm pretty sure I once (maybe even twice) knew what a digamma function is, but it's one of (unfortunately) many concepts that I encountered, but never had any use for, and consequently forgot.

I also second David Roberts's comments, especially the last sentence. It seems that someone has been going through Anixx's old posts and flagging them as spam. That sort of behavior, if it's a result of the current discussion, strikes me as childish and inappropriate.
15.
@Anixx:

To pull a number out of thin air which others may disagree with - I would say you should say something about any term that less than 10% of the readers on MathOverflow know about vaguely. (Someone might say 5% and someone else might say 25%, but I hope I have at least the right order of magnitude.)

You don't need to define sine - 100% of mathematicians know it. You don't need to define Gamma - 90% of mathematicians know it and I am the weird exception (and I know I am the exception). Polygamma is something that fewer than 5% of people on Mathoverflow know, so you need to at least link to a definition for it and say something about it. I would write "The polygamma function (link to definition on Wikipedia) is a special function related to the usual Gamma function." (Undoubtedly you can be a little more informative and accurate than I since I know nothing about it.)

Please keep in mind that Mathoverflow is an international forum and mathematics education varies quite a bit between different countries. Unless you are from a country with a very heterogeneous education system like the United States, there are probably subjects that all mathematicians in your country know very well but most mathematicians in other countries know not at all.

We will all be sometimes wrong in our guesses about what others know. However, I hope that we do recognize our mistakes and correct them when given feedback.

What I hope you do understand from this thread is that, while you may come from a part of the world where classical analysis and particularly special functions is a subject almost all mathematicians know, most of us do not come from such a part of the world and know quite little about special functions.
16.
My comments are intended to be about the subject of the thread and not any particular post or why in particular people have closed Anixx's post.

I think an important point that is perhaps not being made is that there is a difference between a post that could/should be closed and a post that is not user friendly. While many MO users are likely unfamiliar with special functions, those who are most likely to be able to answer a question on special functions will know what the polygamma function is. So I don't think it is right to close or downvote a question just for not defining terms which can be found in, say Wikipedia. For example, there are questions on this site concerning things like $(\infty,1)$-toposes which do not define what is an $(\infty,1)$-topos. I must confess that although I know what is a Grothendieck topos and an elementary topos, I have no idea what is an $(\infty,1)$-topos. But I do not feel that one should oblige all people asking questions about them to define them or even link to the nlab page. On the other hand, I rarely will upvote a question that I can't understand.
17.
Could we just be more accommodating on both sides here? If someone (like Anixx) posts a question, that person should provide definitions of only terms and notation that this person believes is not standard or widely understood. On the other hand, if despite that person's initial effort, people post comments asking for clarification (politely, I hope), then the original questioner should accomodate these requests and provide (politely and without complaint) the definitions requested. The fact is that no matter what you think "all mathematicians" should know, every mathematician I know of, no matter what the stature, has gaps in knowledge. And if they are interested enough in your question to ask for clarification, then you should accomodate their request and welcome their desire to help.
18.
I'm going to bow out of this thread soon, because I have several more interesting things to do. If it were up to me, the solution would be as follows:

* MO users don't close/downvote questions which appear to be reasonably formulated, even if a lot of things are undefined. Leaving comments to ask what things mean is fine.

* We see whether anyone shows up to answer/upvote Annix's questions/answers. If so, those people can have a productive conversation with Annix. If not, then he clearly should provide more detail. I am betting on the latter, but I see no need to force the issue.
19.

+1 David S

• CommentAuthormarkvs
• CommentTimeMar 7th 2012

@Bill: polygamma is a close relative of Hurwitz zeta function which is a (straightforward) generalization of Riemann zeta. If you also do not know what Riemann zeta is, you may indeed want to consider returning a part of your salary to the state of Texas.
20.
markvs, your second sentence is way, way out of line.
21.

I'm pretty sure the State of Texas takes a chunk of Bill's salary in taxes, so that's already taken care of Mark. IMO it's best to contribute positively, rather than throwing out passive-aggressive insults. Mathematics is a big world and what is standard and important doctrine to one person might be something another person never pays attention to, and might even consider not very useful or even insightful. This isn't unusual. We live in a world where very different perspectives on the same thing are usually accepted, and even appreciated, I hope.

22.
@Ryan Budney : Actually, there is no state income tax in Texas :)
23.

Ah. Well I imagine they get part of his salary back in things like sales taxes, then. Bill should go out and buy a beer. If my Googling skills are true, Texas taxes that at a 6.25% rate.

24.
@markvs: it's not only that your second statement is completely out of line, but also the argument implied by the first part is flawed. It's partly because everyone knows about the zeta function and its importance, there are hundreds of "relatives of its generalisations" introduced, and not knowing of some of them is far from being embarrassing.
• CommentAuthorAnixx
• CommentTimeMar 8th 2012

Digamma is not "generalization of zeta function". It is logarithmic derivative of Gamma function, i.e. Г'(x)/Г(х), similarly to what cotangent is for sine.
25.

Well, I am in Canada right now, and it collected taxes on the wine I bought in town last night (as I got tired of drinking beer at the Banff Center), so Texas will have to wait a few days to collect more taxes from me. As for Mark, he is jealous because my title is longer than his.*

26.

*Mark was just joking with someone (me) he knows, just as I am jesting now with him.

27.
Well, if this was a friendly joke between two friends, then I'm quite relieved. I definitely often react incorrectly to exchanges like this.
• CommentAuthorMariano
• CommentTimeMar 8th 2012 edited

(May I suggest that friendly interchanges in the form of unfriendly interchanges be more clearly marked as such? The forum is read by many people who have absolutely no way of knowing which is which —or who knows who— and it would be good to make it clearer for them! This does reflect on the image of MO as a friendly place.)

• CommentAuthormarkvs
• CommentTimeMar 8th 2012

@Mariano: May I gently suggest that you improve your humor recognition skills? Total lack of sense of humor (plus occasional lack of the sense of shame, plus bullying newcommers) reflect on the image of MO as an intelligent place.
28.

@markvs: I did not know you and Mariano are friends!

• CommentAuthorMariano
• CommentTimeMar 8th 2012

Oh well.

29.
Mariano, that should be "who knows whom". Your English teachers may want to consider returning a part of their salary to their employers.
30.

I second Mariano's parenthetical remark. I have no idea who markvs is outside meta.MO and my impression of him (her?) here is that he (she?) does not hold back. So my reading of the initial exchange was that markvs was being quite rude to Bill. I'm pleased to hear that that is not the case.

I'm quite prepared to be told that I have no sense of humor, since I don't. I'm also, to return a little closer to the main topic, clearly an absolute ignoramus as I have no idea what the "polygamma" function is (is it to do with Greek parrots?) and only a hazy idea of the gamma function. I am not ashamed of this ignorance whatsoever: if I ever need to know, I'll look them up on the nLab. If they're not there, then clearly they aren't important.

As for whether or not the question should have explained these terms, that's up to the original poster. But someone who comes here to ask a question should remember that they come as a supplicant: they are trying to get someone to do something for them with no real reward. So when told, "It would help if you did X", it might be worth considering whether or not X will help get the question answered, or if not doing X will throw up such a furore that the actual question gets lost in the cross-fire. Then it might be worth just doing X instead of standing on some principle.

• CommentAuthorAnixx
• CommentTimeMar 9th 2012

@Andrew Stacey, I already understood that this site is not for mathematical analysis, but for theoretical algebra. Any math analysis questions here should be posted with great caution because most people do not understand them and even less can help. This is a pity really because I know no mathematical analysis related Q&A site for mathematicians.
31.
@Anixx: The question isn't tagged properly. It's not merely "calculus" or "analysis", but rather "special functions". That lack of proper tagging is one reason that you are getting to the wrong audience.
Beyond that, "are these identities interesting?" doesn't seem to be the best of questions. If you want the question reopenned, may I suggest that instead of creating drama on meta (thus decreasing your de facto "reputation" on the MO community), that you edit the question, its tags, and its title, so as to make it a question which experts in the field would be likely to find, would understand, and would want to answer.
32.

Regarding Anixx's last comment, and assuming for sake of argument that questions in algebraic functional analysis do not count as "math analysis questions",

http://mathoverflow.net/questions/85266/many-brownian-motions-moving-together/85297#85297

http://mathoverflow.net/questions/19079/what-is-the-status-of-the-gauss-circle-problem/20923#20923

http://mathoverflow.net/questions/43462/existence-of-a-smooth-function-with-nowhere-converging-taylor-series-at-every-poi/81479#81479   http://mathoverflow.net/questions/84958/is-sum-limits-n0-infty-xn-sqrtn-positive/85048#85048

just by skimming over the contributions of one expert.

33.
• CommentAuthorAnixx
• CommentTimeMar 9th 2012 edited

@Yemon Choi there are math analysis questions here but they are rare and often closed. Even following the last link of yours it can be seen that that question was voted to be closed. Only if such questions receive good answers people who do not understand it starting to revert their close/delete votes. So the probability of a question to be closed mostly depends on where there is a good answer from an expert in the field rather than on the question itself.
• CommentAuthorAnixx
• CommentTimeMar 9th 2012 edited

@ Vladimir Dotsenko I do not expect answers from those who are not interested or do not understand the question (particularly those who do not know what is Gamma function). The problem with my last question was that it was advertised in meta as an example of "spam overflow" with which everybody is invited to help to manage. On the other hand I really noticed that people complained that a question lacks definitions of rather basic terms as I view them.
34.
@Anixx: the point is that even for people who could in principle answer your questions, the way they worded might be user-unfriendly enough to ignore or downvote them. (It does happen to me with some questions close to my immediate areas of expertise!) I think that if your question is about the logarithmic derivatives of Gamma function, then it would not take much of your effort to say so, instead of using the term digamma. Quite a few people I asked in offline conversations yesterday, some of them good analysts, did not know / care for the terms digamma/polygamma, while their knowledge might be quite enough to answer your question once the definitions are unwrapped. You can keep telling yourself and the world that these are basic terms, but I doubt anyone would find this sort of reaction on your side productive, let alone pleasant.
• CommentAuthorAnixx
• CommentTimeMar 9th 2012 edited

> Quite a few people I asked in offline conversations yesterday, some of them good analysts, did not know / care for the terms digamma/polygamma, while their knowledge might be quite enough to answer your question once the definitions are unwrapped.

Sorry, I do not believe in this. Even if so, the questions and answers are much better formulated with digamma/polygamma (how would you describe polygamma in terms of only Gamma function by the way?). I never in my life seen "Г'(x)/Г(x)" instead of digamma function used. Similarly i wonder how would you discribe polygamma of negative/fractional order with only "Г" symbol.
• CommentAuthorMariano
• CommentTimeMar 9th 2012

It may be the shameless bully in me, but I do not think there is any point to this thread anymore.

35.
@Anixx: I personally still do not quite care what polygamma means. If only digamma can be described in terms of gamma, and polygamma is something else, then explain that too. If you do not explain, stop wondering why your questions remain unanswered, downvoted and closed: unless you adopt a strategy close to what Daniel Moskovich suggested, nothing is likely to change.
@Mariano: I agree, and vow to stop leaving comments, even if Anixx posts something that looks as though a response is expected.