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:
- Is it true that currently, hypergeometric series are generally not taught often at universities across the world?
- If so, why is this the case?
Best Answer
[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)