# Why are the hypergeometric functions called “hypergeometric”?

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?

• mathworld.wolfram.com/HypergeometricFunction.html – Karn Watcharasupat Apr 12 '18 at 8:42
• – GEdgar Apr 12 '18 at 12:50
• In Graham, Knuth, and Patashnik's Concrete Mathematics they have a comment when introducing the hypergeometric series on the etymology of the name in relation to the geometric series. I do not have the book in front of me right now. Maybe someone else can check. – KCd Apr 12 '18 at 13:21
• On Page 206 (Second Edition) they write: "... is called hypergeometric series because it includes the geometric series as a very special case." – p4sch Apr 12 '18 at 16:14

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 $$\frac{1}{1-x} = {}_1F_0(1;;x)\,.$$
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.
• So the etymology is "geometric sequence" $\to$ "geometric distribution" $\to$ "hypergeometric distribution" $\to$ "hypergeometric function"? I found it slightly disorienting that the answer started in the middle of that chain rather than at one of the ends -- but not so disorienting that I'd downvote. – Henning Makholm Apr 12 '18 at 12:59