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Assuming $\lvert x\lvert<1$ and $0<a<c$, the following formula holds true $$F(a,b,c;x)=\sum_{n=0}^{+\infty} \frac{(a)_n(b)_n}{(c)_n(1)_n} x^n=\frac{\Gamma ( c )}{\Gamma(a)\Gamma(c-a)}\int_0^1 t^{a-1}(1-t)^{c-a-1}(1-xt)^{-b} dt\, , $$ where $(d)_n:=d(d+1)\cdots(d+n-1)$ for every $d$ and every $n\in \mathbb N$.

To whom is due this formula (Euler?) and where precisely can I find its first appearance in the classical literature?

Thanks in advance.

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    $\begingroup$ Kummer proved it in 1836 in his paper "Uber die hypergeometrische Reihe ..." Journal de Crelle 15 (1836) 127--172 ( & 27). It is not found in the paper of Gauss. Later Riemann obtained it in other way. $\endgroup$
    – juan
    Commented Feb 10, 2016 at 12:32

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The following paper discusses the hypergeometric functions and its history in mathematics (for the integral representation cf. p. 26 and references given there):

  • J. Dutka. The early history of the hypergeometric function. Archive for History of Exact Sciences Vol. 31, No. 1 (1984), pp. 15-34

JSTOR link

According to Dutka, Euler came close to the integral representation in his investigation of the hypergeometric differential equation. Some integral representation was found by Legendre in 1816. Apparently the first appearance of the now-known integral representation is in the dissertation of P.C.O Vorsselman de Heer in 1833 (three years before Kummer).

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  • $\begingroup$ @wendt I an still learning something new every day. $\endgroup$
    – juan
    Commented Feb 11, 2016 at 19:53

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