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Recently, I've become more and more interested in hypergeometric series. One of the things that struck me was how it provides a unified framework for many simpler functions. For instance, we have

$$ \log(1+x) = x\ {_2F_1}\left(1,1;2;-x\right) ;$$ $$ \sin^{-1}(x) = x\ {_2F_1}\left(1/2,1/2;3/2;x^{2}\right) ;$$ $$e^{x} = \lim_{b \to \infty} \ {_2F_1}\left(1,b;1;x/b\right), $$ and many more similar identities.

When I saw this for the first time, I was intrigued. Yet at the same time I was also surprised because I hadn't seen it before, and I studied mathematics at a university. I did a quick check, and it seems only a handful of universities in the Netherlands teach hypergeometric series, usually during the late stages of the bachelor's degree or during the master's degree. I am not sure about other countries, but I suspect they're not very often part of the curriculum over there either.

Considering the subject's potential to unify many functions and ideas in analysis, I think it could be useful to learn more about this topic. So my question is twofold:

  1. Is it true that currently, hypergeometric series are generally not taught often at universities across the world?
  2. If so, why is this the case?
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    $\begingroup$ What would you remove from the university curriculum, Max, to make room for hypergeometric series? $\endgroup$ – Gerry Myerson Apr 25 at 12:12
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    $\begingroup$ @GerryMyerson Hmmm I'm not sure I would really want to replace another course with one on hypergeometric series if it were up to me. Perhaps I would try to offer it as an additional elective, and make students (especially those more interested in analysis) aware of this course and its applications. In my former university, we were encouraged to follow a course on measure theory. I found the course quite abstract and difficult, and I think I would have opted for a course on either hypergeometric series, Fourier analysis, or both if those options would've been available at the time. $\endgroup$ – Max Muller Apr 25 at 12:30
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    $\begingroup$ There are many important subjects which are not covered by the modern standard university curricula. Another example is elliptic functions. $\endgroup$ – Alexandre Eremenko Apr 25 at 12:59
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    $\begingroup$ Take advanced physics classes. $\endgroup$ – Tom Copeland Apr 25 at 13:05
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    $\begingroup$ What is the point of unifying $\log(1+x)$ and $\sin^{-1}(x)$ and $e^x$ in this way? Usually we call something a unified framework when it allows a simpler approach or one with fewer special cases. But I don't see any property of these three functions which is most easily proved by expressing them in terms of the hypergeometric function. $\endgroup$ – Matt F. Apr 27 at 1:21
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[Q1] Gert Heckman from Nijmegen University teaches a course on hypergeometric functions (here are the lecture notes, first taught at Tsinghua Univ.).

[Q2] In the foreword, Heckman hints at why this topic is not more popular. Citing Dyson$^\ast$ he notes "two extreme archetypes of mathematicians. On the one hand there are the birds. Like eagles they fly high up in the air and have a magnificient view of the mathematical landscape. They see the great analogies in mathematics for example between geometry and number theory or geometry and mathematical physics. On the other hand there are the frogs. They live down in the mud, and are eager to spot some precious stone hidden under the dirt that the birds might miss."
The study of hypergeometric series is for frogs.


$^\ast$ Some mathematicians are birds, others are frogs. Birds fly high in the air and survey broad vistas of mathematics out to the far horizon. They delight in concepts that unify our thinking and bring together diverse problems from different parts of the landscape. Frogs live in the mud below and see only the flowers that grow nearby. They delight in the details of particular objects, and they solve problems one at a time. I happen to be a frog, but many of my best friends are birds. Mathematics needs both birds and frogs. Mathematics is rich and beautiful because birds give it broad visions and frogs give it intricate details. Mathematics is both great art and important science, because it combines generality of concepts with depth of structures. It is stupid to claim that birds are better than frogs because they see farther, or that frogs are better than birds because they see deeper. The world of mathematics is both broad and deep, and we need birds and frogs working together to explore it. (F.J. Dyson, 2008)
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    $\begingroup$ Thank you for your answer. So I guess what you're also implying is that you think “frog-like” mathematical subjects are currently somewhat less fashionable than “bird-like” subjects? $\endgroup$ – Max Muller Apr 25 at 14:31
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    $\begingroup$ I think so; hypergeometric series have the flavor of 19th century math, it seems like a fully explored territory; although there are new twists to the topic, for example see these lectures on multivariate hypergeometric functions. $\endgroup$ – Carlo Beenakker Apr 25 at 15:12
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    $\begingroup$ I find your latter explanation about people feeling like the main results were already obtained in the 19th century more persuasive than the birds vs. frogs distinction, for one because there are many areas of mathematics today that prize continued technical work on a single subject, and for another because hypergeometric functions have a great number of connections to different fields of math, perhaps most birdishly as test cases for the geometric Langlands program. $\endgroup$ – Will Sawin Apr 26 at 20:33
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    $\begingroup$ @Carlo Beenakker : A lot, if not all, of undergrad, if not earlier, maths is "fully explored territory". That doesn't mean it is not useful. Calculus, linear algebra, much less earlier subjects like Euclidean geometry which is decidable on a computer. All are useful in doing more sophisticated, higher-level things. $\endgroup$ – The_Sympathizer Apr 28 at 3:50
  • $\begingroup$ I sympathize with The Sympathizer. Achieving depth in any topic in any discipline involves understanding its history. Math is organic. To understand it deeply,, you must understand its evolution, just as for human nature, contrary to what a mystic platonist might have us believe. Even flighty categorists start with examples of 19th century set theory or a thousands-of-year-old algorithm a cooking recipe, to introduce their sub-discipline/culture. HGFs are part of the ongoing evolution and could even be used as a hub for exploring and motivating diverse topics stressed in college. $\endgroup$ – Tom Copeland Apr 29 at 21:20
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I think you are correct that a university course on hypergeometric functions is rare. Instead, a course on ordinary differential equations may include a section on hypergeometric functions, as an illustration of the Frobenius method of series solutions for linear ODEs. The useful properties of hypergeometric functions all follow from this.

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    $\begingroup$ Note that "useful properties of [special case] all follow from [general case]" is often used to avoid teaching the special case which may sometimes be a dis-service to students. $\endgroup$ – Kapil Apr 26 at 3:53
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    $\begingroup$ Undergraduate differential equations courses often have a section on series expansions, but this is hardly ever more than an exercise in the algebra of calculating coefficients. The subject is rarely taught properly, with proofs and all that. The natural place to do it would be the second semester of a complex analysis course, but that is seldom done for reasons I never understood. $\endgroup$ – Michael Renardy May 2 at 1:34
  • $\begingroup$ @michaelRenardy Well, having a second complex analysis course at all is rare enough, at least in the US. $\endgroup$ – Kevin Arlin May 2 at 23:54
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Some physics departments (e. g., in Russia) traditionally have a course called "Special functions", that would include hypergeometric functions as well as e. g. Bessel functions, classical orthogonal polynomials and such. I also think we did touch them at mathematics department, perhaps in a Mathematical physics course.

The reason they are often skipped in Math education is, as others have pointed out, that they mostly arise as solutions of a special type of second-order ODE, so the natural place for them would be an ODE course. However, the properties of these ODE manifest most naturally in a complex variable — in fact, the class of ODEs in question are those which have at most three regular singular points on the Riemann sphere; the case of two such points leads to elementary functions; so in this sense, the hypergeometric ODE is the simplest one not solved in elementary functions.

The ODE courses tend to avoid relying on complex variables, since the students might not have this prerequisite, and even if the ODE course is advanced, there are so many nice and important topics it can cover instead, both on the theoretical and the applied side.

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    $\begingroup$ Complex analysis is a natural domain of study for electrical/mechanical engineers and physicists and in presenting trigonometry. Neglecting it in courses on diff eqs? Sounds like pre-Euler textbooks are being used. (I realize most high school calculus textbooks neglect Euler's formula. That's a real disservice also.) $\endgroup$ – Tom Copeland Apr 26 at 22:49
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    $\begingroup$ "Euler's formula" is a many-valued expression. $\endgroup$ – Gerry Myerson Apr 27 at 6:00
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    $\begingroup$ @TomCopeland, I partially agree with the sentiment, but take a typical book on ODE, for example the Arnold's. To include hypergeometric equations, you would need to 'neglect' one of the topics covered in the book; which one would you sacrifice? $\endgroup$ – Kostya_I Apr 27 at 6:27
  • $\begingroup$ When it comes to organized religion and the current state of affairs in academe, I find it best to follow the proverbial mom's advice. $\endgroup$ – Tom Copeland Apr 28 at 21:55
  • $\begingroup$ @TomCopeland For the record, Euler's identity $e^{i \theta} = \cos \theta + i \sin \theta$ is featured in every high school pre-calculus textbook I've come across, typically in tandem with de Moivre's identity. The HS calc textbook I checked just now contains exercises on proving $\frac{1}{2i}(e^{ix} - e^{-ix}) = \sin x$ and $\frac{1}{2}(e^{ix} + e^{-ix}) = \cos x$. There's no particular reason to spend a lot of time on complex-variable identities in a course on real-variable calculus, especially a beginner-level course. $\endgroup$ – Rivers McForge May 2 at 2:25
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The hypergeometric functions arise naturally in the study of second order differential equations and, therefore, courses in mathematical physics. See "A Catalogue of Sturm-Liouville differential equations" by Everitt and "PDEs, ODEs, Analytic Continuation, Special Functions, Sturm–Liouville Problems and All That" by Burgess. Since they are related to second order equations, there is a connection to the Schwarzian derivative and all that entails. For a history of the HGFs, see Linear Differential Equations and Group Theory from Riemann to Poincare by Jeremy Gray.

The confluent hypergeometric functions include the families of Laguerre polynomials from which the Hermite polynomials can be constructed. These appear not only in quantum physics, probability theory, and analysis involving the heat, harmonic oscillator, and Schrodinger equations but also combinatorics. They also have associated ladder operators which form instances of a Graves/Lie/Heisenberg/Weyl algebra. Operators associated with the Laguerre polynomials can be related to a Witt-Lie algebra as well. OEIS A131758 has associations among the confluent hypergeometric functions and basic functions and number sequences arising in number theory and topology/characteristic classes. Special functions of the hypergeometric type are rife in group/representation theory and operational calculus, explored by Miller, Gilmore, Vilenkin, Carlitz, el-Salaam, Rota, Askey, Wilson, among others. Peruse also the book Hypergeometric Functions, My Love by Yoshida.

From the book A = B by Petkovsek, Wilf, and Zeilberger on hypergeometric identities important in general combinatorics and analysis of algorithms:

"Hypergeometric series are very important in mathematics. Many of the familiar functions of analysis are hypergeometric. These include the exponential, logarithmic, trigonometric, binomial, and Bessel functions, along with the classical orthogonal polynomial sequences of Legendre, Chebyshev, Laguerre, Hermite, etc.

It is important to recognize when a given series is hypergeometric, if it is, because the general theory of hypergeometric functions is very powerful, and we may gain a lot of insight into a function that concerns us by first recognizing that it is hypergeometric, then identifying precisely which hypergeometric function it is, and finally by using known results about such functions."

So, hypergeometric functions are studied by mathematicians and physicists but rarely in their full generality in classes—a symptom of specialization, motivated/rationalized more often than not by economics and an assembly-line, industrial mindset towards education.

(Dyson's dichotomy is needlessly polarizing. Dyson, who was proud to point out he had no Ph.D., unified Feynman's and Schwinger's approaches to QED and was subsequently awarded tenure at Princeton's IAS. He was perhaps the proverbial frog with wings. The greats like Riemann and John von Neumann break the mold, being shape-shifters morphing between the eagle and the jaguar, prowling both the skies and the hidden jungle beneath the canopy.)


Edit (4/28/20)

Addressing a question by Matt F and the continuing relevance of HGFs to modern mathematics, I'd like to point out relations among generalized differential operators and the confluent HGFs illustrated in the MO-Q&A "Pochhammer symbol of a differential and hypergeometric polynomial" and the paper "Contributions to the hypergeometric function theory of Heckman and Opdam: sharp estimates, Schwartz space, heat kernel" by Schapira (2018).

I first glimpsed the relation between the Pochhammer symbol/rising factorial and the diff op reps of the Kummer HGFs in the books on generalized functions by Gelfand and Shilov. Members of a family of KHGFs and families of KHGFs are related via conjugation of generalized diff ops, i.e., Rodriques-style formulas. For integer parameters, such diff op reps are related to important combinatoric constructs, such as the Dobinski formula as discussed by Rota in relation to the Stirling numbers of the second kind. See also, e.g., "Notes on the Rodrigues formulas for two kinds of the Chebyshev polynomials" by Feng Qi, Da-Wei Niu, Dongkyu Lim (2020). The importance of Laguerre, Hermite, Legendre, Chebyshev, etc. in diverse areas of math and physics ranging from analytic number theory to combinatorics to string theory and topology is well-documented.

I'm currently revisiting the lit on the connections between extremum principles and differential equations, particularly, the heat equation. The paper by Schapira cited above illustrates the continuing interest in these and associated topics, based on the HGFs presented in "Root systems and hypergeometric functions I" by Heckman and Opdam (1987).

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  • $\begingroup$ I went to look up the articles, and added two links I found, though I'm not sure if they are authoritative. \\ What is the significance of the final parenthetical? In particular, what bird? $\endgroup$ – LSpice Apr 26 at 20:04
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    $\begingroup$ @LSpice, counterpoint to Carlo's allusion to Dyson's pernicious metaphor/meme on the birds and the frogs. $\endgroup$ – Tom Copeland Apr 26 at 20:13
  • $\begingroup$ Ah: @CarloBeenakker's allusion. MO currently shows me your answer at the top, so I hadn't seen that. $\endgroup$ – LSpice Apr 26 at 20:24
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    $\begingroup$ Do you, or do the authors of A=B, have a good example of an insight from recognizing that a certain function is hypergeometric, and which results about hypergeometric functions would be used? $\endgroup$ – Matt F. Apr 28 at 14:18
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    $\begingroup$ (cont) Also, e.g., with the HGS rep it is easy to develop Mellin transforms of HGFs/HGSs as in the Mellin-Barnes contour integral rep and, consequently, related generalized Dobinski formulas. $\endgroup$ – Tom Copeland Apr 29 at 16:02
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Hypergeometric functions and series are maybe not taught in pure mathematics courses but they are often taught in more advanced physics courses.

If you see the appendices of some of the books on basic theoretical physics by Landau and Lifshitz, for example, hypergeometric series should be mentioned in there (the book on non-relativistic quantum mechanics definitely has some stuff on this).

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I find some banal answers to this question convincing:

  • Mathematicians prefer to focus on functions of one or two variables.
  • The power series for $_2F_1$ requires an unusual number-theoretic function (the Pochhammer symbol).
  • The position of $_2$ before $F_1$ makes the hypergeometric function almost uniquely awkward to typeset or to read about.

As a comparison: Are the Pearson distributions not taught often at universities nowadays, and if so, why? They provide a unified framework for many simpler probability distributions, e.g. \begin{align} B(\alpha,\beta) &= \operatorname{Pearson}_\text I(2-\alpha-\beta, \alpha-1, 1, -1, 0)\\ \Gamma(\alpha,\beta) &= \operatorname{Pearson}_\text{III}(1, \beta-\alpha\beta, 0, \beta, 0)\\ N(\mu,\sigma) &= \operatorname{Pearson}_\text{III}(1, -\mu, 0, 0, \sigma^2). \end{align} The subscripts here indicate the distribution type, which can also be calculated from the parameters.

(There are plenty of parallels with the hypergeometric function: The arcsin and exponential distributions are subcases of $B$ and $\Gamma$. The five parameters for the Pearson distribution are projectively invariant, so they have the same degrees of freedom as $_2F_1$. And Pearson was apprarently motivated to construct these distributions by analyzing the hypergeometric distribution, whose cdf uses $_3F_2$, and the corresponding differential equation.)

I think the answer is clear: The Pearson distributions are not taught often. While they unify other distributions, it is rare that this unification is useful for proving anything.

Occasionally the closest fit to some data is a Pearson IV distribution. Even then, we have so little intuition about the distribution and so little sense of a mechanism that would generate it, that it's usually better not to use such a fit.

In short: the Pearson distributions are so ugly that most people prefer to avoid them when they can. The hypergeometric functions may be similar.

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  • $\begingroup$ If cumbersome typesetting were to dissuade mathematicians from subjects, why have dozens of people written books on continued fractions? $\endgroup$ – Duncan W Apr 28 at 15:49
  • $\begingroup$ Typesetting $_2F_1$ is not unusually cumbersome. The typesetting problem is that there's no way to make it look good, at least to my eyes. E.g. when I first saw this question, I tried to edit it to make it look better -- when I couldn't, I realized that this was probably part of the larger issue. $\endgroup$ – Matt F. Apr 28 at 16:49

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