# Where does the word “hypergeometric” come from?

A friend of mine asked me how the "hypergeometric distribution" got its name.

My guess is that an answer to this question will explain what is "geometric" about the distribution, and also why mathematicians/statisticians seem to be so fond of the prefix "hyper-".

According to http://jeff560.tripod.com/h.html:

The term HYPERGEOMETRIC (to describe a particular differential equation) is due to Johann Friedrich Pfaff (1765-1825) (Kline, page 489).

The term HYPERGEOMETRIC CURVE is found in the title "De curva hypergeometrica hac aequatione expressa y=1*2*3*...*x" by Leonhard Euler. The paper was presented in 1768 and published in 1769 in Novi Commentarii academiae scientiarum Petropolitanae.

HYPERGEOMETRIC DISTRIBUTION appears in the title of H. T. Gonin, "The use of factorial moments in the treatment of the hypergeometric distribution and in tests for regression," Philosophical Magazine, 7, Ser. 21, 215-226 (1936).

The name is relatively recent but the distribution first appears as the solution to Problem IV of Huygens’s De Ratiociniis in Ludo Aleae (1657, p. 12). Several people, besides Huygens, solved the problem and James Bernoulli and de Moivre gave solutions for the general case. See Hald (1990, pp. 201-2). At the end of the 19th century Karl Pearson wrote a paper in which he considered fitting the distribution (given by the "hypergeometrical series") to data: "On Certain Properties of the Hypergeometrical Series, and on the Fitting of such Series to Observation Polygons in the Theory of Chance," Philosophical Magazine, 47, (1899), 236-246. [John Aldrich]

HYPERGEOMETRIC SERIES. According to Geschichte der Elementar-Mathematik by Karl Fink, Wallis and Euler used this term for the series in which the quotient of any term divided by the preceding is an integral linear function of the index, and J. F. Pfaff proposed the term for the general series in which the quotient of any term divided by the preceding is a function of the index.

The 1816 translation of Lacroix's Differential and Integral Calculus has: "These series, in which the number of factors increases from term to term, have been designated by Euler ... hypergeometrical series" (OED2).

So, it seems hypergeometric series and the corresponding differential equations came first. The hypergeometric distribution got its name (much later) from the fact that its probability generating function involves a hypergeometric function: $$E[s^X] = {\frac {{N-R\choose n}{\mbox{_2F_1}(-R,-n;\,N-R-n+1;\,s)}}{{N \choose n}}}$$

• Thanks for the answer! However, it seems now that the natural question is "what was so "hypergeometric" about this particular differential equation"? – Fredrik Meyer Feb 5 '13 at 16:07
• It's really from the series first, I think. A geometric series $\sum_n a_n$ has $a_{n+1}/a_n$ constant. A hypergeometric series has $a_{n+1}/a_n$ a rational function of $n$. – Robert Israel Feb 5 '13 at 18:22