Talk:Generalized hypergeometric function

Notation

Notation not straight yet - I'll have to try to sort it out, since the actual conventions are not what one would guess a priori.

Charles Matthews 18:17, 16 Nov 2003 (UTC)

This page seems very weak to me. It would be better to start with the most general definition (n upper parameters and m lower parameters), explain the connection with recurrences, and finally move on to famous special cases and identities. Also, the traditional notation is rather ugly, incorporating as it does the number of parameters into the name of the symbol. Would it make sense to use the much nicer Graham/Knuth/Patashnik notation from Concrete Mathematics for the main presentation, and mention the traditional notation in an aside? Gdr 15:05, 2004 Aug 2 (UTC)

No, I don't think it is right to start with the most general definition. And I don't think the notation that occurs in the classical works should be junked just on the basis of one graduate textbook. There is clearly a huge amount that could be added - the subject being a couple of centuries old - but please, can we have some sort of sensible gradient at the start rather than plunging into F_mⁿ kind of stuff?

But by all means add something about the GKP notation and its advantages.

Charles Matthews 15:30, 2 Aug 2004 (UTC)

Hypergeometric series

Hi Linas,

thank you for your welcoming note and your encouragement. I did not find a button to send an e-mail, so I hope you read this. Just a comment on the edit in "Hypergeometric series" before my edits. In fact, I made this, too, but before I created the account, so it's anonymous. I still think, $z$ should be ommitted from these formulas, but as you reverted this, I should maybe give some additional explanation:

If we have a series $\sum _{n=0}^{\infty }a_{n}z^{n}$, by convention, the coeffitients of the series are $a_{n}$, while the terms or summands are $a_{n}z^{n}$.

Now consider the the exponential series

$1+z+{\frac {1}{2}}z^{2}+{\frac {1}{6}}z^{3}+\cdots$.

Then the ratio of two successive terms is given by ${\frac {z}{n}}$ while the ratio of two successive coeffitients is given by ${\frac {1}{n}}$. We are talking about coefficients here, aren't we?

Maybe I missunderstood an important point about these functions; if you insist in the necessity of $z$ in these formulas, please tell my your reasoning.

Regards, Thomas (Thomas Bliem 10:33, 15 November 2005 (UTC)).

Right, except that that is an unusual definition of a series. If $\{a_{n}\}$ is a sequence, then the series is $\sum _{n}a_{n}$ (there is no assumed z^n in the definition of a series, in general). Rather, the article is explicitly trying to discuss the ratio a_n = tilda-p/tilda-q, from which it factors out a factor of z/n leaving the ratio p/q which is now the traditional hypergeometric series. That is, the ratio of two successive coefficients is tilda-p/tilda-q = (z/n)(p/q). The exponential function has p=q=1. The point of the exercise is to demonstrate why there is a factor of z^n/n! that is traditionally removed (and not just z^n alone, or 1/n! alone). Without this, the intro section doesn't hang together.

Anyway, thank you very much for your concern; we really like conscientious editors on wikipedia. If you can come up with some alternate way of stating the above, I suppose that's OK, as long as the article is better as a result.

Oh, and in general, we should have the conversation on the talk page of the article. at elast on a matter such as this. linas 15:07, 15 November 2005 (UTC)

I guess the point you are working from is that if {a_n} is a seuqnce, then the corresponding geometric series would be Sigma_n a_n z^n, and now all that is needed is the explanation of the hyper- prefix ... I guess either development would work. linas 15:20, 15 November 2005 (UTC)

Like Thomas said, if you have a series \sum_n a_n, then the individual a_n are called "terms". If you have a power series \sum_n a_n x^n, then the terms are a_n x^n and the coefficients are a_n. I change the first sentence from

"a hypergeometric series is the sum of a sequence of terms in which the ratios of successive coefficients k is a rational function of k."

to

"a hypergeometric series is a power series in which the ratios of successive coefficients k is a rational function of k."

Is this clearer? -- Jitse Niesen (talk) 16:01, 15 November 2005 (UTC)

Yes, except that the starting point is not a power series. The ending point is a power series. The starting point is a sequence a_n the ratio of whose terms is a rational function of n. That is, a_{n+1} / a_n = tilde-P(n) / tilde-Q(n) where tilde-P(n) and tilde-Q(n) are "any" polynomials. In particular, the very first step is to remove the leading constant: that is, to write tilde-P(n) / tilde-Q(n) = const. \cdot n^m + O(n^{m-1}) for some integer m. One then identifies the leading const. with z (a ha!). After removal of this leading const, one then gets the rest of this stuff with the usual normal form/normalization. The factor of z just takes you from "any" rational function, to the suitable normalized rational function. See where I'm coming from?

Of course, if one defines a polynomial (or rational function) to be something for which the const. in front of the leading term is already set to unity, then one has to introduce the z in some other way (e.g. by declaring it to be a power series by fiat). It doesn't matter much which way you do it, except that maybe there's a tiny bit of insight gained from knowing that the z can be factored into/pulled out of the leading const. of the rational function. Its kind of "projective" in this way. linas 22:37, 15 November 2005 (UTC)

Okay, I think I see what you mean, but I don't see why you would do that. Why is the leading constant identified with z? -- Jitse Niesen (talk) 12:13, 16 November 2005 (UTC)

Well, start with the first term, and multiply them out (convert the ratio of terms into terms). The series will then have (const.)^n in the n'th term. Right? So identify const. with z, or merge it into z, or just call it z, whatever. linas 15:58, 16 November 2005 (UTC)

I don't understand why this has to be confusing. I don't see why there have to be tildes in the notation, for example. Just say in words that the old P, Q can be replaced. Words are better for almost all humans, you know, rather than notation. Charles Matthews 12:17, 16 November 2005 (UTC)

I agonized over this when I first wrote it. Its bad style to re-use the same notation for two different (but related) items; naive readers are all too easily mislead. I was concerned that some reader would equate the P in the first paragraph to the P in the second, and get confused. With some agony and displeasure, I picked tilde-P as a passable solution, if not a very good one.

As to formulas vs. words: I like both: converting words into formulas is error-prone; however, formulas are needed for calculations. For general reading, words are great. However, as I often calculate, I need to have accurate formulas: there's nothing quite like wasting an afternoon of calculations because the initial formula had some missing term or some subtle mis-statement. linas 15:58, 16 November 2005 (UTC)

As far as I know, you don't have to repeat letters until you have used 26. This kind of tangle in expression usually means a rewrite is needed, anyway. And I don't think we should be writing introductory material based so closely on personal (good or bad) experiences. Charles Matthews 16:06, 16 November 2005 (UTC)

Hmm, I was trying to stick to "P" for polynomial; but I don't much care. I have no plans for a re-write; I wrote the current draft, its someone-else's turn. And surely, the "personal experience" of calculating the wrong thing is broadly and regularly shared by all, beginner or sophisticate? I can't possibly be the only one running around making mistakes! Seriously, I think its just fine to belabour "obvious" points, if it can get some college kid to understand and get interested. WP has had a lot of "too technical" debates recently. linas 03:45, 17 November 2005 (UTC)

Level of generality

Well, I wonder why I bothered writing above that I didn't think we should start with the general one-variable case. Charles Matthews 11:50, 18 November 2005 (UTC)

revert of recent edits

Hi, I just reverted some well-meaning recent edits by User:Ben Standeven mostly because the fell into te same trap discussed above, on circa 15 Nov 2005. linas 02:11, 1 March 2006 (UTC)

• I think I get it now; but while the distinction between a hypergeometric series and a hypergeometric function is important, the old article didn't really explain it. So I've switched back to my version for that section. I've left the intro as is, though. Ben Standeven 19:38, 6 March 2006 (UTC)

The generality of the introduction is excessive; I've been arguing a point like this at Talk:General linear group, and I'll argue it here. And, Linas, the tildes are just as bad an idea as they ever were. Charles Matthews 19:56, 6 March 2006 (UTC)

I removed the tildes. I agree in principle about excess generality in math articles; I am not sure what to do about it here. I presume that a simpler structure would be:

1. give series formula for 2F1. A few words about analytic continuation; a discussion of 2F1-related topics.
2. give series formula for mFn.
3. explain why its called "hypergeometric" (viz, what the current "intro" does).
4. rest of article as it currently stands.

Basically, the only change would be to add a leading section on 2F1 and another on mFn. Would that work? linas 15:22, 7 March 2006 (UTC)

Yes, with the lead section only about 2F1, it wouldn't be so tough. Charles Matthews 15:43, 7 March 2006 (UTC)

I started making these edits. linas 14:29, 8 March 2006 (UTC)

I'm done, for now. This exercise makes clear how pathetic this article is. Oh well. linas 15:28, 8 March 2006 (UTC)

Relationship to varieties?

I started making the edits above, and it doesn't help that I've been reading about varieties and schemes recently. It occurs to me: 2F1 is related to the elliptic curves (in this strange way I haven't yet mastered; it follows through the diff eq for the j-invariant). Does this relationship generalize to varieties in general? A quick skim of the index of a few books indicates that there is no discussion of such things. linas 14:29, 8 March 2006 (UTC)

Maybe not. The relation is through the j-invariant. Do varieties in general have j-invariants? I guess not, I guess that the j-invariant is due to the richness of the moduli space. Should I guess that the higher-dimensional structures don't have interesting moduli spaces, possibly due to some rigidity theorem? Who knows. And so I add another question on my infinite list of unanswered questions. linas 16:49, 8 March 2006 (UTC)

There are other good moduli spaces - for example, for K3 surfaces. I think you're looking for the Gauss-Manin connection, the differential equation for the 'periods', though. Charles Matthews 21:15, 8 March 2006 (UTC)

Odd sentence

"In a certain sense, the situation can be likened to using a computer to perform addition and multiplication; the actual value of the resulting number is in a sense less important than the various patterns that emerge; and so it is with hypergeometric identities as well." -- what does that mean? McKay 14:23, 11 June 2006 (UTC)

Examples?

It would be nice to see some actual examples of how other functions may be represented by special cases of the hypergeomtric function and also some notes on the domain of convergence. I'm finding the mathworld article (http://mathworld.wolfram.com/HypergeometricFunction.html) a more useful reference. Farmhouse121 03:14, 13 August 2006 (UTC)

rising factorial or falling factorial?

In the paragraph "The series ₂F₁", the sentence "where $(a)_{n}=a(a+1)(a+2)\ldots (a+n-1)$ is the rising factorial", is it wrong?

According to the article Pochhammer symbol, $(a)_{n}$ seems to be a symbol of a falling factorial.

——Nussknacker胡桃夹子^.^tell me... 19:52, 29 September 2006 (UTC)

The symbol $(a)_{n}$ is sometimes used for either one, and that is the source of the problem. It's not wrong, it's just inconsistent with the Wikipedia Pochhammer article. I think you have a good point, it should be made consistent. PAR 14:13, 30 September 2006 (UTC)

Can someone provide references to some of the assertions here, such as that the Gamma function and the elliptic integrals are cases of hypergoemetric functions/series? These do not seem to appear on the corresponding pages and look suspicious to me. 82.230.82.110 20:41, 12 December 2006 (UTC)

Check out Abramowitz and Stegun, pages 509 Eq. 13.6.10 and 510 Eq. 13.6.28 for the Gamma function, and page 591 Eqs. 17.3.9-10 for the elliptic integrals. PAR 23:28, 12 December 2006 (UTC)

I concur that elliptic integrals can be expressed as 2F1s (it's a special case of Euler's integral representation). However, the tables do not give a formula for the Gamma function, but rather for the incomplete Gamma function. Indeed, the Gamma function itself is not a hypergeometric function as defined in the article; in order to express Gamma in hypergeometric terms, the argument of Gamma must appear in the parameters of the hypergeometric series, not the variable. 209.125.235.244 18:25, 14 April 2007 (UTC)

Elementary functions

I've recently added a little bit of stuff to this article – I was trying to address some of the concerns (usefulness of the article, "Pochhammer symbols", etc) raised on this talk page. In particular, I added some of the simpler elementary functions that can be expressed as particular instances of the hypergeometric function. I had an ulterior motive for doing this. I intend to tie part of this article into a new article Continued fraction of Gauss which I've been intending to write for some time now.

One other thing. The article says "Many other special cases are listed in the Category:Special hypergeometric functions." But that's just a list of additional articles. I checked out several of those yesterday, and found no mention of the hypergeometric series in those associated articles. Shouldn't each of the articles in that category contain an explicit reference to this article? Or at least show how that function can be expressed as an instance of a hypergeometric function? DavidCBryant 15:49, 30 May 2007 (UTC)

Another special case

Does anybody know of another representation of $\,_{0}F_{1}(;a;c/z)$? I have an integral that involves this hypergeometric function, an exponential, etc., and can't perform the integral.

Any help would be greatly appreciated. I realise that this isn't so much a discussion, but perhaps the answer (if there is one) might be worth putting on the main page. Thanks! --Wainson (talk) 20:56, 29 August 2008 (UTC)

It is related to a Bessel function (see Bessel function#Relation to hypergeometric series). In Abramowitz and Stegun (http://www.math.sfu.ca/~cbm/aands/page_355.htm) there's lots of formulae with integrals of Bessels with exponentials etc. Maybe you can find an answer there. Greets, David 09:09, 17 October 2008 (UTC)

Hypergeometric identities

There are two articles relating to the term "hypergeometric identities", namely List of hypergeometric identities and Hypergeometric identities. The first article deals with identities involving hypergeometric functions and the second with identities involving the terms of hypergeometric series. These are related but different topics and having the virtually same name for both is confusing. I'm changing the destination of the link in the identities section to the first article, and removing the remark about algorithms for proving them since it seems to be referring to the second article. Is there better terminology to avoid this confusion in the future?--RDBury (talk) 06:45, 13 February 2009 (UTC)

Arithmetic-geometric mean

The following

$\,_{2}F_{1}\left({\frac {1}{2}},{\frac {1}{2}};1;k^{2}\right)={\frac {1}{agm(1-k,1+k)}}={\frac {2}{\pi }}K(k)\,\!$

Where

• ${agm}\,\!$ = the arithmetic-geometric mean
• $K(k)\,\!$ = the complete elliptic integral of the first kind

can be inferred from

• Elliptic_integral#Complete_elliptic_integral_of_the_first_kind

-- Ac44ck (talk) 16:48, 21 March 2010 (UTC)

Why generalized hypergeometric functions?

This article seems to do a good job of describing the generalized hypergeometric function and its properties.

But it leaves this reader with a burning question: Why is this class of functions worth studying?

For many other mathematical entities, their extremely special and beautiful properties alone make the answer to this question clear. But for the class of generalized hypergeometric function, its properties seem to be about what one might expect if we consider the level of complexity of the definition of these functions. (Which is not too simple and not too complex.) The properties of these functions — as described in the article — are not particularly ugly but they do not strike me as particularly beautiful or special, either.

I am probably missing some important insights. Quite possibly many other readers will wonder the same thing: Why study generalized hypergeometric functions? I am not very knowledgeable in this area, so I hope someone who is knowledgeable can fill in this omission.Daqu (talk) 18:42, 26 October 2016 (UTC)

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Needs Reference

"It can be shown that the series converges absolutely at z = 1 if..." needs a reference. Skewray (talk) 21:19, 30 July 2023 (UTC)

Also, "Analytic continuation can be employed for larger values of z." What does this mean? Skewray (talk) 21:19, 30 July 2023 (UTC)