Are hypergeometric series not taught often at universities nowadays, and if so, why?

Question

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?

Answer 1

[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)}

Answer 2

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.

Answer 3

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.

Answer 4

Added April 13, 2023: (Start)

Taking a proactive rather than reactive stance, two articles, in addition to the ones below in my response and comments, could be used to advocate for an elective course on hypergeometric functions that would serve as a hub to introduce advanced undergraduates who wish to pursue careers in advanced research to a wide variety of subfields in math, physics, science, and engineering (including the dismal science):

"Three lectures on hypergeometric functions" (2006) by Eduardo Cattani (noted in one of Carlo's comments)

Excerpt:

There has been a great revival of interest in the study of hypergeometric functions in the last two decades. Indeed, a search for the title word hypergeometric in the MathSciNet database yields 3181 articles of which 1530 have been published since 1990! This newfound interest comes from the connections between hypergeometric functions and many areas of mathematics such as representation theory, algebraic geometry and Hodge theory, combinatorics, D-modules, number theory, mirror symmetry, etc.

"Hypergeometric Functions: From One Scalar Variable to Several Matrix Arguments, in Statistics and Beyond" (2016) by T. Pham-Gia and Dinh Ngoc Thanh

Excerpt:

8. Hypergeometric Functions in Neighboring Domains

8.1. Algebraic topology, Algebraic K-Theory, Algebraic Geometry

8.1.1. Integral representations

8.1.2. Single Integral representation

8.1.3. A-Hypergeometric functions

8.2. Hypergeometric integrals in Conformal Field theory, Homology and Cohomology

8.3. Algebraic functions and roots of equations

8.4. Economics, Quantitative Economics and Econometrics

8.5. Random matrices in Theoretical Physics

(The last circles back to free moments and cumulants in free probability theory, which are related to generalized multivariate Catalan and Narayana / parking polynomials, which are ... , which are ..., which are ... .)

(End)

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-Salam, Rota, Roman, 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 / limited resources and an assembly-line, industrial mindset towards education.

(Aside: 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 et al. in "From sets to functions: Three elementary examples" in relation to the Stirling numbers of the first and second kinds. 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).

Answer 5

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).

Edit: If you were curious, the hypergeometric functions do turn up in physics like quite a lot, see for example page 12 of this article, where $_2F_1$ makes an appearance.

Answer 6

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.