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?
Best Answer
In the 19th century, a lot of efforts were made in order to solve the general quintic equation $x^5+a_4x^4 +a_3x^3 +a_2x^2 +a_1x +a_0$ using special functions. It turns out that the roots of this equation are expressible in terms of hypergeometric series. To wit, one possibility is by first reducing the number of parameters, to the form $x^5-x-t=0$. Then a Lagrange inversion argument essentially gives a root $$ z=t {}_4 F_3(\frac15,\frac25,\frac35,\frac45,\frac12,\frac34,\frac54,\frac{5^5}{4^4}t^4)=t+t^5+10\frac{ t^9}{2!}+15\cdot 14 \frac{t^{13}}{3!}+\ldots $$