TITLE: Why are hypergeometric series important and do they have a geometric or heuristic motivation?
QUESTION [42 upvotes]: Apart from telling that the hypergeometric functions (or series) are the solutions to the (essentially unique?) fuchsian equation on the Riemann sphere with 3 "regular singular points", the wikipedia article doesn't illuminate much about why this kind of special functions should form such a natural topic in mathematics (and in fact have been throughout 19th century).
Simply:

What are hypergeometric series really, and why they should be (or have been in the past centuries) important/interesting?

REPLY [13 votes]: Hypergeometric functions arise as matrix coefficients of representations of Lie groups. 
This is my formal answer, but I will also describe very informally why I believe this is an answer to your question. (By this I roughly mean: why this might have triggered interest of 19th century mathematicians who didn't have the language to be aware that this is what they were looking at. However: I do no not know a thing about the history of the subject so this is just a mathematical remark from a modern perspective.)
Interesting geometric spaces tend to have a rich group of symmetries (either because that is what makes them interesting (e.g. the circle), or because the symmetry is the only thing that gives us any grip on the object (e.g. spacetime)). Also studying a space through the functions on it has proved to be a powerful way to do mathematics. Hence representations of Lie groups on vector spaces of functions arise naturally. 
Now it turns out that the matrix coefficient functions (of smaller representations) really give you a grip on these rather huge representations. The canonical example is when the group is compact and the interesting space on which the functions live is the group itself. In this case the Peter-Weyl theorem states that matrix coefficient functions form a basis. (Which at least makes it kind of credible that they are also useful for understanding functions on other compact homogeneous spaces.)
Now not everything is compact. (This is relevant here as Gauss' hypergeometric function appears as a matrix coefficient of an $SL(2, \mathbb{R})$-representation. (Incidentally this was the answer to prof. Koornwinder's question at my own thesis defense, see my comment to the original post above.)) However, also in this case matrix coefficient functions help you understand general representations on function spaces. For instance, in the infinite dimensional case (for non-compact groups irreducible representations need not be finite dimensional) it is not immediately clear (to say the least) that a given Lie algebra representation integrates to group representation. One way to make this work (given some extra conditions) is to first construct the matrix coefficients of the hypothesized group representation and then construct the actual representation from that (see for instance [1]). To me this always felt a bit like proving the existence of the Yeti from its footprint, but the difference is that it actually works.
Final remark: this is just the first part of JME's answer in disguise since in the setting of a Lie group acting on a space of functions on a homogeneous space (for that Lie-group), the differential equations JME is talking about come from the action of the Lie algebra.
[1] v.d. Ban: Induced representations and the Langlands classification [http://www.staff.science.uu.nl/~ban00101/manus/edinb.pdf]