TITLE: Are hypergeometric series not taught often at universities nowadays, and if so, why?
QUESTION [43 upvotes]: 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?

REPLY [4 votes]: 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.