Talk:Generalized hypergeometric function

message: 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)
replies: text: 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 Fmn kind of stuff?
But by all means add something about the GKP notation and its advantages.
Charles Matthews 15:30, 2 Aug 2004 (UTC)
thread_title: Notation

message: 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)
thread_title: Level of generality

message: 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)
replies: text: 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)
thread_title: Relationship to varieties?

message: "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)
thread_title:  Odd sentence 

message: 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)
thread_title:  Examples? 

message: 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)
replies: text: 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)

text: 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)
replies: text:  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)
thread_title:  [[rising factorial]] or [[falling factorial]]? 

message: 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)
thread_title:  Elementary functions 

message: Does anybody know of another representation of  \,_0F_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)
replies: text: 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)
thread_title:  Another special case 

message: 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)
thread_title:  Hypergeometric identities 

message: The following 
:
\,_2F_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)
thread_title:  Arithmetic-geometric mean 

message: 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)
thread_title:  Why generalized hypergeometric functions? 

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thread_title:  External links modified 

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