I was wondering where the term "hypergeometric" for the hypergeometric function $_2F_1(a, b; c; z)$ comes from. Wikipedia says that the term was coined by J. Wallis, but I couldn't find any (mathematical) reason why these functions are anything like hyper-geometric.
Does anyone know where this comes from?
On
This is the reason I believe: a geometric sequence is defined as a sequence where the ratio of two consecutive numbers is a constant $$ \frac{a_{n+1}}{a_n} = x\,. $$ You can generalize this notion by assuming the ratio to be any rational function of $n$ instead $$ \frac{a_{n+1}}{a_n} = \frac{P(n)}{Q(n)}\,. $$ Any such rational function can be factorized and rewritten as $$ \frac{P(n)}{Q(n)} = \frac{(n+a_1)\cdots(n+a_p)}{(n+b_1)\cdots(n+b_q)(n+1)}x\,. $$ If $Q(n)$ doesn't have an $n+1$ factor we can always say, for example, $a_p=1$, so we don't lose generality. With this parametrization it's easy to check that $$ \sum_{n=0}^\infty a_n = {}_pF_q(a_1,\ldots, a_p;b_1,\ldots, b_q;x)\,. $$ Hypergeometric then suggests that this is a generalization of the geometric sequence. More precisely $$ \frac{1}{1-x} = {}_1F_0(1;;x)\,. $$
Source: http://mathworld.wolfram.com/HypergeometricFunction.html
The probability mass function of the hypergeometric distribution (related to the binomial distribution and the geometric distribution) is given by a ratio of products of binomial coefficients, just like the coefficients of the MacLaurin series of a hypergeometric function. This is my bet on the reason for picking the adjective hypergeometric for describing the $\phantom{}_p F_q$ functions.