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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?

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    $\begingroup$ mathworld.wolfram.com/HypergeometricFunction.html $\endgroup$ – Karn Watcharasupat Apr 12 '18 at 8:42
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    $\begingroup$ see also math.stackexchange.com/q/295258/442 $\endgroup$ – GEdgar Apr 12 '18 at 12:50
  • $\begingroup$ 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. $\endgroup$ – KCd Apr 12 '18 at 13:21
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    $\begingroup$ On Page 206 (Second Edition) they write: "... is called hypergeometric series because it includes the geometric series as a very special case." $\endgroup$ – p4sch Apr 12 '18 at 16:14
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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

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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.

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  • $\begingroup$ It would be nice to know if the downvote is due to the fact that I am wrong, or something else. $\endgroup$ – Jack D'Aurizio Apr 12 '18 at 12:48
  • $\begingroup$ 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. $\endgroup$ – Henning Makholm Apr 12 '18 at 12:59
  • $\begingroup$ @HenningMakholm: I am not sure that is the historical chain of events, but is sounds likely, doesn't it? $\endgroup$ – Jack D'Aurizio Apr 12 '18 at 13:16

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